Area of a Region Between Two Curves

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1 6 CHAPTER 7 Applictions of Integrtion Section 7 Are of Region Between Two Curves Find the re of region etween two curves using integrtion Find the re of region etween intersecting curves using integrtion Descrie integrtion s n ccumultion process Are of Region Between Two Curves Region etween two curves g f With few modifictions ou cn etend the ppliction of definite integrls from the re of region under curve to the re of region etween two curves Consider two functions f nd g tht re continuous on the intervl, If, s in Figure 7, the grphs of oth f nd g lie ove the -is, nd the grph of g lies elow the grph of f, ou cn geometricll interpret the re of the region etween the grphs s the re of the region under the grph of g sutrcted from the re of the region under the grph of f, s shown in Figure 7 = = Figure 7 g g g f f f Are of region etween f nd g f g d Figure 7 Are of region under f f d Are of region under g g d Animtion f( i ) Figure 7 Representtive rectngle Height: f( i ) g( i ) Width: i g( i ) g f To verif the resonleness of the result shown in Figure 7, ou cn prtition the intervl, into n suintervls, ech of width Then, s shown in Figure 7, sketch representtive rectngle of width nd height f i g i, where i is in the ith intervl The re of this representtive rectngle is A i heightwidth f i g i B dding the res of the n rectngles nd tking the limit s n, ou otin lim n n f i g i i Becuse f nd g re continuous on,, f g is lso continuous on, nd the limit eists So, the re of the given region is Are lim n n f i g i i f g d

2 SECTION 7 Are of Region Between Two Curves 7 Are of Region Between Two Curves If f nd g re continuous on, nd g f for ll in,, then the re of the region ounded the grphs of f nd g nd the verticl lines nd is A f g d In Figure 7, the grphs of f nd g re shown ove the -is This, however, is not necessr The sme integrnd f g cn e used s long s f nd g re continuous nd g f for ll in the intervl, This result is summrized grphicll in Figure 7 f() g() (, f()) f g f() g() (, f()) f g (, g()) (, g()) NOTE The height of representtive rectngle is f g regrdless of the reltive position of the -is, s shown in Figure 7 Figure 7 Representtive rectngles re used throughout this chpter in vrious pplictions of integrtion A verticl rectngle of width implies integrtion with respect to, wheres horizontl rectngle of width implies integrtion with respect to EXAMPLE Finding the Are of Region Between Two Curves f() = + (, f()) (, g()) g() = Region ounded the grph of f, the grph of g,, nd Figure 75 Find the re of the region ounded the grphs of,,, nd Solution Let g nd f Then g f for ll in,, s shown in Figure 75 So, the re of the representtive rectngle is A f g nd the re of the region is A f g d d 7 6 Editle Grph Tr It Eplortion A Eplortion B

3 8 CHAPTER 7 Applictions of Integrtion Are of Region Between Intersecting Curves In Emple, the grphs of f nd g do not intersect, nd the vlues of nd re given eplicitl A more common prolem involves the re of region ounded two intersecting grphs, where the vlues of nd must e clculted EXAMPLE A Region Ling Between Two Intersecting Grphs Find the re of the region ounded the grphs of f nd g (, f()) g() = f() = (, g()) Region ounded the grph of f nd the grph of g Figure 76 Solution In Figure 76, notice tht the grphs of f nd g hve two points of intersection To find the -coordintes of these points, set f nd g equl to ech other nd solve for or Set f equl to g Write in generl form Fctor Solve for So, nd Becuse g f for ll in the intervl,, the representtive rectngle hs n re of A f g nd the re of the region is A d 9 Editle Grph Tr It EXAMPLE Eplortion A A Region Ling Between Two Intersecting Grphs The sine nd cosine curves intersect infinitel mn times, ounding regions of equl res, s shown in Figure 77 Find the re of one of these regions π (, g()) (, f()) π π g() = cos f() = sin One of the regions ounded the grphs of the sine nd cosine functions Figure 77 Solution sin cos sin cos tn or 5, Set f equl to g Divide ech side cos Trigonometric identit Solve for So, nd 5 Becuse sin cos for ll in the intervl, 5, the re of the region is 5 5 A sin cos d cos sin Editle Grph Tr It Eplortion A

4 SECTION 7 Are of Region Between Two Curves 9 If two curves intersect t more thn two points, then to find the re of the region etween the curves, ou must find ll points of intersection nd check to see which curve is ove the other in ech intervl determined these points EXAMPLE Curves Tht Intersect t More Thn Two Points Find the re of the region etween the grphs of g f nd g() f() f() g() 6 (, ) (, ) 6 (, 8) 8 g() = + f() = On,, g f, nd on,, f g Figure 78 Editle Grph Solution Begin setting f nd g equl to ech other nd solving for This ields the -vlues t ech point of intersection of the two grphs,, 6 6 Set f equl to g Write in generl form Fctor Solve for So, the two grphs intersect when,, nd In Figure 78, notice tht g f on the intervl, However, the two grphs switch t the origin, nd f g on the intervl, So, ou need two integrls one for the intervl, nd one for the intervl, A f g d g f d d d Tr It Eplortion A Open Eplortion NOTE In Emple, notice tht ou otin n incorrect result if ou integrte from to Such integrtion produces f g d d If the grph of function of is oundr of region, it is often convenient to use representtive rectngles tht re horizontl nd find the re integrting with respect to In generl, to determine the re etween two curves, ou cn use A top curve ottom curve d Verticl rectngles in vrile A right curve left curve d Horizontl rectngles in vrile where, nd, re either djcent points of intersection of the two curves involved or points on the specified oundr lines Technolog

5 5 CHAPTER 7 Applictions of Integrtion EXAMPLE 5 Horizontl Representtive Rectngles Find the re of the region ounded the grphs of nd Solution Consider g nd These two curves intersect when nd, s shown in Figure 79 Becuse f g on this intervl, ou hve So, the re is A d f A g f d 8 9 Tr It Eplortion A f() = + (, ) = (, ) = (, ) g() = Horizontl rectngles (integrtion with respect to ) Figure 79 Editle Grph (, ) = Verticl rectngles (integrtion with respect to ) Figure 7 In Emple 5, notice tht integrting with respect to ou need onl one integrl If ou hd integrted with respect to, ou would hve needed two integrls ecuse the upper oundr would hve chnged t, s shown in Figure 7 A d d 6 d d 9

6 SECTION 7 Are of Region Between Two Curves 5 Integrtion s n Accumultion Process In this section, the integrtion formul for the re etween two curves ws developed using rectngle s the representtive element For ech new ppliction in the remining sections of this chpter, n pproprite representtive element will e constructed using preclculus formuls ou lred know Ech integrtion formul will then e otined summing or ccumulting these representtive elements Known preclculus formul Representtive element New integrtion formul For emple, in this section the re formul ws developed s follows A heightwidth A f g A f g d EXAMPLE 6 Descriing Integrtion s n Accumultion Process Find the re of the region ounded the grph of nd the -is Descrie the integrtion s n ccumultion process Solution The re of the region is given A d You cn think of the integrtion s n ccumultion of the res of the rectngles formed s the representtive rectngle slides from to, s shown in Figure A d A d 5 A d A d 9 Figure 7 A d Tr It Eplortion A

7 5 CHAPTER 7 Applictions of Integrtion Eercises for Section 7 The smol Click on Click on indictes n eercise in which ou re instructed to use grphing technolog or smolic computer lger sstem to view the complete solution of the eercise to print n enlrged cop of the grph In Eercises 6, set up the definite integrl tht gives the re of the region f 6 f g f g g g f In Eercises nd, find the re of the region integrting () with respect to nd () with respect to f g f g Think Aout It In Eercises 5 nd 6, determine which vlue est pproimtes the re of the region ounded the grphs of f nd g (Mke our selection on the sis of sketch of the region nd not performing n clcultions) In Eercises 7, the integrnd of the definite integrl is difference of two functions Sketch the grph of ech function nd shde the region whose re is represented the integrl 7 8 g d sec d g d d f f 5 5 f 6 f g g f g d sec cos d f g 5 6 () () (c) (d) (e) 8 () () 6 (c) (d) (e) In Eercises 7, sketch the region ounded the grphs of the lgeric functions nd find the re of the region f, f,,,, 8 8,,, 8 f, g f, g f, g f, g,,,,, 5 f, g f, g f, g f, g f, g,, f g g, g, 6 f,,, g,,

8 SECTION 7 Are of Region Between Two Curves 5 In Eercises, () use grphing utilit to grph the region ounded the grphs of the equtions, () find the re of the region, nd (c) use the integrtion cpilities of the grphing utilit to verif our results In Eercises 8, sketch the region ounded the grphs of the functions, nd find the re of the region In Eercises 9 5, () use grphing utilit to grph the region ounded the grphs of the equtions, () find the re of the region, nd (c) use the integrtion cpilities of the grphing utilit to verif our results In Eercises 5 56, () use grphing utilit to grph the region ounded the grphs of the equtions, () eplin wh the re of the region is difficult to find hnd, nd (c) use the integrtion cpilities of the grphing utilit to pproimte the re to four deciml plces f, g f, g,,, f, g f, g f, g f 6,,,,,, f sin, g tn, f sin, g cos, 6 f cos, g cos, f sec tn f e,, f, g f sin sin,, f sin cos,, < f e,, g ln,, 5,,,, cos e,,,, g, In Eercises 57 6, find the ccumultion function F Then evlute F t ech vlue of the independent vrile nd grphicll show the re given ech vlue of F 57 F t dt () F () F (c) F6 58 F t dt () F () F (c) F6 59 F cos () F () F (c) F d 6 F e d () F () F (c) F In Eercises 6 6, use integrtion to find the re of the figure hving the given vertices 6,,, 6, 6, 6,,,,, c 6 6,,,,,,,,,,,,,, 65 Numericl Integrtion Estimte the surfce re of the golf green using () the Trpezoidl Rule nd () Simpson s Rule ft ft 66 Numericl Integrtion Estimte the surfce re of the oi spill using () the Trpezoidl Rule nd () Simpson s Rule mi ft 5 mi ft mi 5 ft In Eercises 67 7, set up nd evlute the definite integrl tht gives the re of the region ounded the grph of the function nd the tngent line to the grph t the given point 67 f,, 68,, mi ft mi ft 5 mi 5 ft 69 f 7, Writing Aout Concepts 6 ft 5 mi mi 6 ft,,, 7 The grphs of nd intersect t three points However, the re etween the curves cn e found single integrl Eplin wh this is so, nd write n integrl for this re

9 5 CHAPTER 7 Applictions of Integrtion Writing Aout Concepts (continued) 7 The re of the region ounded the grphs of nd cnnot e found the single integrl d Eplin wh this is so Use smmetr to write single integrl tht does represent the re 7 A college grdute hs two jo offers The strting slr for ech is $,, nd fter 8 ers of service ech will p $5, The slr increse for ech offer is shown in the figure From strictl monetr viewpoint, which is the etter offer? Eplin Slr (in dollrs) 6, 5,,,,, In Eercises 75 nd 76, find such tht the line divides the region ounded the grphs of the two equtions into two regions of equl re 75 9, 76 In Eercises 77 nd 78, find such tht the line divides the region ounded the grphs of the equtions into two regions of equl re 77,, 78, In Eercises 79 nd 8, evlute the limit nd sketch the grph of the region whose re is represented the limit 79 lim i i, where i in nd n n i 8 lim i, where i in nd n n S i Offer Offer 6 8 Yer Figure for 7 Figure for 7 t Deficit (in illions of dollrs) 7 A stte legislture is deting two proposls for eliminting the nnul udget deficits the er The rte of decrese of the deficits for ech proposl is shown in the figure From the viewpoint of minimizing the cumultive stte deficit, which is the etter proposl? Eplin 9, Revenue In Eercises 8 nd 8, two models nd re given for revenue (in illions of dollrs per er) for lrge corportion The model R gives projected nnul revenues from to 5, with t corresponding to, nd R gives projected revenues if there is decrese in the rte of growth of corporte sles over the period Approimte the totl reduction in revenue if corporte sles re ctull closer to the model R 6 5 D Proposl R Proposl t 6 Yer R 8 R 8 R 7 6t t 7 58t R 7 5t 8 Modeling Dt The tle shows the totl receipts R nd tot ependitures E for the Old-Age nd Survivors Insurnce Trus Fund (Socil Securit Trust Fund) in illions of dollrs The time t is given in ers, with t corresponding to 99 (Source: Socil Securit Administrtion) t R E t R E () Use grphing utilit to fit n eponentil model to the dt for receipts Plot the dt nd grph the model () Use grphing utilit to fit n eponentil model to the dt for ependitures Plot the dt nd grph the model (c) If the models re ssumed to e true for the ers through 7, use integrtion to pproimte the surplus revenue generted during those ers (d) Will the models found in prts () nd () intersect? Eplin Bsed on our nswer nd news reports out the fund, will these models e ccurte for long-term nlsis? 8 Lorenz Curve Economists use Lorenz curves to illustrte the distriution of income in countr A Lorenz curve, f represents the ctul income distriution in the countr In this model, represents percents of fmilies in the countr nd represents percents of totl income The model represents countr in which ech fmil hs the sme income The re etween these two models, where, indictes countr s income inequlit The tle lists percents of income for selected percents of fmilies in countr () Use grphing utilit to find qudrtic model for the Lorenz curve () Plot the dt nd grph the model R 7 t t (c) Grph the model How does this model compre with the model in prt ()? (d) Use the integrtion cpilities of grphing utilit to pproimte the income inequlit

10 SECTION 7 Are of Region Between Two Curves Profit The chief finncil officer of compn reports tht profits for the pst fiscl er were $89, The officer predicts tht profits for the net 5 ers will grow t continuous nnul rte somewhere etween % nd 5% Estimte the cumultive difference in totl profit over the 5 ers sed on the predicted rnge of growth rtes 86 Are The shded region in the figure consists of ll points whose distnces from the center of the squre re less thn their distnces from the edges of the squre Find the re of the region Figure for 86 Figure for Mechnicl Design The surfce of mchine prt is the region etween the grphs of nd 8 k (see figure) () Find k if the prol is tngent to the grph of () Find the re of the surfce of the mchine prt 88 Building Design Concrete sections for new uilding hve the dimensions (in meters) nd shpe shown in the figure (55, ) = 6 5 () Find the re of the fce of the section superimposed on the rectngulr coordinte sstem () Find the volume of concrete in one of the sections multipling the re in prt () meters (c) One cuic meter of concrete weighs 5 pounds Find the weight of the section 89 Building Design To decrese the weight nd to id in the hrdening process, the concrete sections in Eercise 88 often re not solid Rework Eercise 88 to llow for clindricl openings such s those shown in the figure m 8 m 5 + Rottle Grph m 5 6 = 5 (55, ) True or Flse? In Eercises 9 9, determine whether the sttement is true or flse If it is flse, eplin wh or give n emple tht shows it is flse 9 If the re of the region ounded the grphs of f nd g is, then the re of the region ounded the grphs of h f C nd k g C is lso 9 If f g d A, then g f d A 9 If the grphs of f nd g intersect midw etween nd, then f g d 9 Are Find the re etween the grph of sin nd the line segments joining the points, nd 7 s 6,, shown in the figure (, ) 6 Figure for 9 Figure for 9 9 Are Let > nd > Show tht the re of the ellipse 7 6 (, is (see figure) ( + Putnm Em Chllenge 95 The horizontl line c intersects the curve in the first qudrnt s shown in the figure Find c so tht the res of the two shded regions re equl = c = This prolem ws composed the Committee on the Putnm Prize Competition The Mthemticl Assocition of Americ All rights reserved = m 6 5 (55, ) = = 5 (55, ) Rottle Grph

11 56 CHAPTER 7 Applictions of Integrtion Section 7 Volume: The Disk Method Find the volume of solid of revolution using the disk method Find the volume of solid of revolution using the wsher method Find the volume of solid with known cross sections The Disk Method In Chpter we mentioned tht re is onl one of the mn pplictions of the definite integrl Another importnt ppliction is its use in finding the volume of three-dimensionl solid In this section ou will stud prticulr tpe of threedimensionl solid one whose cross sections re similr Solids of revolution re used commonl in engineering nd mnufcturing Some emples re les, funnels, pills, ottles, nd pistons, s shown in Figure 7 Rectngle w Rottle Grph Rottle Grph Rottle Grph R Ais of revolution w Disk R Solids of revolution Figure 7 Rottle Grph Rottle Grph Volume of disk: R w Figure 7 Rottle Grph If region in the plne is revolved out line, the resulting solid is solid of revolution, nd the line is clled the is of revolution The simplest such solid is right circulr clinder or disk, which is formed revolving rectngle out n is djcent to one side of the rectngle, s shown in Figure 7 The volume of such disk is Volume of disk re of diskwidth of disk R w where R is the rdius of the disk nd w is the width To see how to use the volume of disk to find the volume of generl solid of revolution, consider solid of revolution formed revolving the plne region in Figure 7 out the indicted is To determine the volume of this solid, consider representtive rectngle in the plne region When this rectngle is revolved out the is of revolution, it genertes representtive disk whose volume is V R Approimting the volume of the solid n such disks of width nd rdius R i produces Volume of solid n R i i n i R i

12 SECTION 7 Volume: The Disk Method 57 Representtive rectngle Ais of revolution Representtive disk Plne region R = = Disk method Figure 7 Solid of revolution Approimtion n disks Rottle Grph This pproimtion ppers to ecome etter nd etter s n So, ou cn define the volume of the solid s Volume of solid lim R d n R i i Schemticll, the disk method looks like this Known Preclculus Formul Volume of disk V R w Representtive Element V R i New Integrtion Formul Solid of revolution V R d A similr formul cn e derived if the is of revolution is verticl The Disk Method To find the volume of solid of revolution with the disk method, use one of the following, s shown in Figure 75 Horizontl Ais of Revolution Volume V R d Verticl Ais of Revolution d Volume V R d c V = π [R()] d d d V = π c [R()] d NOTE In Figure 75, note tht ou cn determine the vrile of integrtion plcing representtive rectngle in the plne region perpendiculr to the is of revolution If the width of the rectngle is, integrte with respect to, nd if the width of the rectngle is, integrte with respect to R() Horizontl is of revolution Figure 75 c R() Verticl is of revolution

13 58 CHAPTER 7 Applictions of Integrtion The simplest ppliction of the disk method involves plne region ounded the grph of f nd the -is If the is of revolution is the -is, the rdius R is simpl f EXAMPLE Using the Disk Method f() = sin Find the volume of the solid formed revolving the region ounded the grph of f sin nd the -is out the -is π Plne region π R() Solution From the representtive rectngle in the upper grph in Figure 76, ou cn see tht the rdius of this solid is R f sin Figure 76 Solid of revolution π So, the volume of the solid of revolution is V R d cos sin d sin d Appl disk method Simplif Integrte Rottle Grph Tr It Eplortion A Eplortion B EXAMPLE Revolving Aout Line Tht Is Not Coordinte Ais Plne region f() = g() = Find the volume of the solid formed revolving the region ounded f R() nd g out the line, s shown in Figure 77 Solid of revolution Ais of revolution Figure 77 f() g() Solution B equting f nd g, ou cn determine tht the two grphs intersect when ± To find the rdius, sutrct g from f R f g Finll, integrte etween nd to find the volume V R d d d 5 Appl disk method Simplif Integrte Rottle Grph Tr It Eplortion A

14 SECTION 7 Volume: The Disk Method 59 w The Wsher Method R r Ais of revolution The disk method cn e etended to cover solids of revolution with holes replcing the representtive disk with representtive wsher The wsher is formed revolving rectngle out n is, s shown in Figure 78 If r nd R re the inner nd outer rdii of the wsher nd w is the width of the wsher, the volume is given Volume of wsher R r w Disk w r R To see how this concept cn e used to find the volume of solid of revolution, consider region ounded n outer rdius R nd n inner rdius r, s shown in Figure 79 If the region is revolved out its is of revolution, the volume of the resulting solid is given V R r d Wsher method Figure 78 Rottle Grph Solid of revolution Note tht the integrl involving the inner rdius represents the volume of the hole nd is sutrcted from the integrl involving the outer rdius R() r() Plne region Solid of revolution with hole = (, ) Figure 79 Rottle Grph R = (, ) = r = Plne region EXAMPLE Using the Wsher Method Find the volume of the solid formed revolving the region ounded the grphs of nd out the -is, s shown in Figure 7 Solution In Figure 7, ou cn see tht the outer nd inner rdii re s follows R r Outer rdius Inner rdius Solid of revolution Figure 7 Solid of revolution Integrting etween nd produces V R r d d d 5 5 Appl wsher method Simplif Integrte Rottle Grph Tr It Eplortion A

15 6 CHAPTER 7 Applictions of Integrtion In ech emple so fr, the is of revolution hs een horizontl nd ou hve integrted with respect to In the net emple, the is of revolution is verticl nd ou integrte with respect to In this emple, ou need two seprte integrls to compute the volume EXAMPLE Integrting with Respect to,two-integrl Cse Find the volume of the solid formed revolving the region ounded the grphs of,,, nd out the -is, s shown in Figure 7 For : R = r = r R (, ) Solid of revolution For : R = r = Figure 7 Plne region Rottle Grph Solution For the region shown in Figure 7, the outer rdius is simpl R There is, however, no convenient formul tht represents the inner rdius When, r, ut when, r is determined the eqution, which implies tht r r,, Using this definition of the inner rdius, ou cn use two integrls to find the volume V d d d d d Appl wsher method Simplif Integrte Note tht the first integrl represents the volume of right circulr clinder of rdius nd height This portion of the volume could hve een determined without using clculus Tr It Eplortion A Eplortion B Figure 7 Generted Mthemtic TECHNOLOGY Some grphing utilities hve the cpilit to generte (or hve uilt-in softwre cple of generting) solid of revolution If ou hve ccess to such utilit, use it to grph some of the solids of revolution descried in this section For instnce, the solid in Emple might pper like tht shown in Figure 7

16 SECTION 7 Volume: The Disk Method 6 EXAMPLE 5 Mnufcturing () R() = 5 = 5 r() = 5 5 Plne region () Figure 7 Solid of revolution Rottle Grph = in 5 5 in A mnufcturer drills hole through the center of metl sphere of rdius 5 inches, s shown in Figure 7() The hole hs rdius of inches Wht is the volume of the resulting metl ring? Solution You cn imgine the ring to e generted segment of the circle whose eqution is 5, s shown in Figure 7() Becuse the rdius of the hole is inches, ou cn let nd solve the eqution 5 to determine tht the limits of integrtion re ± So, the inner nd outer rdii re r nd R 5 nd the volume is given V R r d 6 Solids with Known Cross Sections 56 cuic inches With the disk method, ou cn find the volume of solid hving circulr cross section whose re is A R This method cn e generlized to solids of n shpe, s long s ou know formul for the re of n ritrr cross section Some common cross sections re squres, rectngles, tringles, semicircles, nd trpezoids 5 d 6 d Tr It Eplortion A Open Eplortion Volumes of Solids with Known Cross Sections For cross sections of re A tken perpendiculr to the -is, Volume A d See Figure 7() For cross sections of re A tken perpendiculr to the -is, d Volume A d See Figure 7() c = = = c () Cross sections perpendiculr to -is Figure 7 Rottle Grph = d () Cross sections perpendiculr to -is Rottle Grph

17 6 CHAPTER 7 Applictions of Integrtion EXAMPLE 6 Tringulr Cross Sections Find the volume of the solid shown in Figure 75 The se of the solid is the region ounded the lines Cross sections re equilterl tringles = g() h Figure 76 Rottle Grph f() = g() = + Tringulr se in -plne Figure 75 Are = A() Are of se = B = Rottle Grph = f() h = (h ) h nd The cross sections perpendiculr to the -is re equilterl tringles Solution f, EXAMPLE 7 The se nd re of ech tringulr cross section re s follows Bse Are se A Becuse rnges from to, the volume of the solid is V A d d Tr It An Appliction to Geometr Length of se Are of equilterl tringle Are of cross section Prove tht the volume of prmid with squre se is V hb, where h is the height of the prmid nd B is the re of the se Solution As shown in Figure 76, ou cn intersect the prmid with plne prllel to the se t height to form squre cross section whose sides re of length Using similr tringles, ou cn show tht h h or where is the length of the sides of the se of the prmid So, A h h Integrting etween nd h produces h V h A d h h h h d h ) d g, hh h h h hb h Eplortion A h B Tr It Eplortion A Eplortion B

18 SECTION 7 Volume: The Disk Method 6 Eercises for Section 7 The smol Click on Click on indictes n eercise in which ou re instructed to use grphing technolog or smolic computer lger sstem to view the complete solution of the eercise to print n enlrged cop of the grph In Eercises 6, set up nd evlute the integrl tht gives the volume of the solid formed revolving the region out the -is 9 5, 6 In Eercises 7, set up nd evlute the integrl tht gives the volume of the solid formed revolving the region out the -is , 5 In Eercises, find the volume of the solid generted revolving the region ounded the grphs of the equtions out the given lines () the -is () the -is (c) the line (d) the line 6 () the -is () the -is (c) the line 8 (d) the line, () the -is () the line 6 () the -is () the line In Eercises 5 8, find the volume of the solid generted revolving the region ounded the grphs of the equtions out the line 5,, 6,, 7 8 In Eercises 9, find the volume of the solid generted revolving the region ounded the grphs of the equtions out the line 6 9,,,, 6, 6,,, sec,,,,, 6 6,,,, 6,, 6, 6 In Eercises, find the volume of the solid generted revolving the region ounded the grphs of the equtions out the -is,,,,

19 6 CHAPTER 7 Applictions of Integrtion In Eercises nd, find the volume of the solid generted revolving the region ounded the grphs of the equtions out the -is In Eercises 6, find the volume of the solid generted revolving the region ounded the grphs of the equtions out the -is Verif our results using the integrtion cpilities of grphing utilit 5 6 In Eercises 7, use the integrtion cpilities of grphing utilit to pproimte the volume of the solid generted revolving the region ounded the grphs of the equtions out the -is 7 8 9,,,,,, 8 e,,, e,,,, 5,,,,, 8,, 9,,, sin,,, cos,,, e,,, e e,,, e,,, ln,,, rctn,,, 5, Writing Aout Concepts In Eercises nd, the integrl represents the volume of solid Descrie the solid sin d Think Aout It In Eercises nd, determine which vlue est pproimtes the volume of the solid generted revolving the region ounded the grphs of the equtions out the -is (Mke our selection on the sis of sketch of the solid nd not performing n clcultions) () () 5 (c) (d) 7 (e) rctn,,, () () (c) 5 (d) 6 (e) 5 e,,, d Writing Aout Concepts (continued) 5 A region ounded the prol nd the -is is revolved out the -is A second region ounded the prol nd the -is is revolved out the -is Without integrting, how do the volumes of the two solids compre? Eplin 6 The region in the figure is revolved out the indicted es nd line Order the volumes of the resulting solids from lest to gretest Eplin our resoning () -is () -is (c) If the portion of the line ling in the first qudrnt is revolved out the -is, cone is generted Find the volume of the cone etending from to 6 8 Use the disk method to verif tht the volume of right circulr cone is r h, where r is the rdius of the se nd h is the height 9 Use the disk method to verif tht the volume of sphere is r 5 A sphere of rdius r is cut plne h h < r units ove the equtor Find the volume of the solid (sphericl segment) ove the plne 5 A cone of height H with se of rdius r is cut plne prllel to nd h units ove the se Find the volume of the solid (frustum of cone) elow the plne 5 The region ounded,,, nd is revolved out the -is () Find the vlue of in the intervl, tht divides the solid into two prts of equl volume () Find the vlues of in the intervl, tht divide the solid into three prts of equl volume 5 Volume of Fuel Tnk A tnk on the wing of jet ircrft is formed revolving the region ounded the grph of 8 nd the -is out the -is (see figure) where nd re mesured in meters Find the tnk s volume = Rottle Grph

20 SECTION 7 Volume: The Disk Method 65 5 Volume of L Glss A glss continer cn e modeled revolving the grph of 9, 95, out the -is, where nd re mesured in centimeters Use grphing utilit to grph the function nd find the volume of the continer 55 Find the volume of the solid generted if the upper hlf of the ellipse is revolved out () the -is to form prolte spheroid (shped like footll), nd () the -is to form n olte spheroid (shped like hlf of cnd) Figure for 55() 56 Minimum Volume The rc of Figure for 55() on the intervl, is revolved out the line (see figure) () Find the volume of the resulting solid s function of () Use grphing utilit to grph the function in prt (), nd use the grph to pproimte the vlue of tht minimizes the volume of the solid (c) Use clculus to find the vlue of tht minimizes the volume of the solid, nd compre the result with the nswer to prt () Rottle Grph < 5 Rottle Grph 6 58 Modeling Dt A drftsmn is sked to determine the mount of mteril required to produce mchine prt (see figure in first column) The dimeters d of the prt t equll spced points re listed in the tle The mesurements re listed in centimeters () Use these dt with Simpson s Rule to pproimte the volume of the prt () Use the regression cpilities of grphing utilit to find fourth-degree polnomil through the points representing the rdius of the solid Plot the dt nd grph the model (c) Use grphing utilit to pproimte the definite integr ielding the volume of the prt Compre the result with the nswer to prt () 59 Think Aout It Mtch ech integrl with the solid whose volume it represents, nd give the dimensions of ech solid () Right circulr clinder () Ellipsoid (c) Sphere (d) Right circulr cone (e) Torus (i) d d (iii) (v) h r r r r h d r r d (ii) (iv) h r d d R r R r d 6 Cvlieri s Theorem Prove tht if two solids hve equ ltitudes nd ll plne sections prllel to their ses nd equl distnces from their ses hve equl res, then the solids hve the sme volume (see figure) = R R h Are of R re of R Figure for 56 Figure for 58 Rottle Grph Rottle Grph Rottle Grph 57 Wter Depth in Tnk A tnk on wter tower is sphere of rdius 5 feet Determine the depths of the wter when the tnk is filled to one-fourth nd three-fourths of its totl cpcit (Note: Use the zero or root feture of grphing utilit fter evluting the definite integrl) 6 Find the volume of the solid whose se is ounded the grphs of nd, with the indicted cross sections tken perpendiculr to the -is () Squres () Rectngles of height Rottle Grph Rottle Grph

21 66 CHAPTER 7 Applictions of Integrtion 6 Find the volume of the solid whose se is ounded the circle with the indicted cross sections tken perpendiculr to the -is () Squres () Equilterl tringles In Eercises 67 7, find the volume generted rotting the given region out the specified line 5 R = R R = 5 Rottle Grph (c) Semicircles Rottle Grph (d) Isosceles right tringles 67 R out 68 R out 69 R out 7 R out 7 R out 7 R out 7 R out 7 out 75 The solid shown in the figure hs cross sections ounded the grph of where, () Descrie the cross section when nd () Descrie procedure for pproimting the volume of the solid R Rottle Grph 6 The se of solid is ounded,, nd Find the volume of the solid for ech of the following cross sections (tken perpendiculr to the -is): () squres, () semicircles, (c) equilterl tringles, nd (d) semiellipses whose heights re twice the lengths of their ses 6 Find the volume of the solid of intersection (the solid common to oth) of the two right circulr clinders of rdius r whose es meet t right ngles (see figure) Two intersecting clinders Rottle Grph Solid of intersection FOR FURTHER INFORMATION For more informtion on this prolem, see the rticle Estimting the Volumes of Solid Figures with Curved Surfces Donld Cohen in Mthemtics Techer MthArticle 65 A mnufcturer drills hole through the center of metl sphere of rdius R The hole hs rdius r Find the volume of the resulting ring 66 For the metl sphere in Eercise 65, let R 5 Wht vlue of r will produce ring whose volume is ectl hlf the volume of the sphere? Rottle Grph 76 Two plnes cut right circulr clinder to form wedge One plne is perpendiculr to the is of the clinder nd the second mkes n ngle of degrees with the first (see figure) () Find the volume of the wedge if () Find the volume of the wedge for n ritrr ngle Assuming tht the clinder hs sufficient length, how does the volume of the wedge chnge s increses from to 9? Figure for 76 Figure for () Show tht the volume of the torus shown is given the integrl 8 R () Find the volume of the torus r + = + = + = Rottle Grph Rottle Grph 5 r d, where R > r > R Rottle Grph r

22 SECTION 7 Volume: The Shell Method 67 Section 7 Volume: The Shell Method Find the volume of solid of revolution using the shell method Compre the uses of the disk method nd the shell method The Shell Method h w p Figure 77 Rottle Grph p w Ais of revolution p + w In this section, ou will stud n lterntive method for finding the volume of solid of revolution This method is clled the shell method ecuse it uses clindricl shells A comprison of the dvntges of the disk nd shell methods is given lter in this section To egin, consider representtive rectngle s shown in Figure 77, where w is the width of the rectngle, h is the height of the rectngle, nd p is the distnce etween the is of revolution nd the center of the rectngle When this rectngle is revolved out its is of revolution, it forms clindricl shell (or tue) of thickness w To find the volume of this shell, consider two clinders The rdius of the lrger clinder corresponds to the outer rdius of the shell, nd the rdius of the smller clinder corresponds to the inner rdius of the shell Becuse p is the verge rdius of the shell, ou know the outer rdius is p w nd the inner rdius is p w p w Outer rdius p w Inner rdius So, the volume of the shell is Volume of shell volume of clinder volume of hole p w h p w h phw verge rdiusheightthickness p() d c h() Plne region You cn use this formul to find the volume of solid of revolution Assume tht the plne region in Figure 78 is revolved out line to form the indicted solid If ou consider horizontl rectngle of width, then, s the plne region is revolved out line prllel to the -is, the rectngle genertes representtive shell whose volume is V ph Figure 78 Rottle Grph Ais of revolution Solid of revolution You cn pproimte the volume of the solid n such shells of thickness, height h i, nd verge rdius p i Volume of solid n p i h i i n This pproimtion ppers to ecome etter nd etter s n So, the volume of the solid is Volume of solid lim n p i h i i d ph d c i p i h i

23 68 CHAPTER 7 Applictions of Integrtion The Shell Method To find the volume of solid of revolution with the shell method, use one of the following, s shown in Figure 79 Horizontl Ais of Revolution d Volume V ph d c Verticl Ais of Revolution Volume V ph d d h() h() c p() p() Horizontl is of revolution Figure 79 Verticl is of revolution EXAMPLE Using the Shell Method to Find Volume Find the volume of the solid of revolution formed revolving the region ounded nd the -is out the -is Solution Becuse the is of revolution is verticl, use verticl representtive rectngle, s shown in Figure 7 The width indictes tht is the vrile of integrtion The distnce from the center of the rectngle to the is of revolution is p, nd the height of the rectngle is h = Becuse rnges from to, the volume of the solid is V ph d d d Appl shell method Simplif h() = p() = Ais of revolution Figure 7 (, ) Integrte Tr It Eplortion A

24 SECTION 7 Volume: The Shell Method 69 EXAMPLE Using the Shell Method to Find Volume p() = Figure 7 h() = e = e Ais of revolution Find the volume of the solid of revolution formed revolving the region ounded the grph of e nd the -is out the -is Solution Becuse the is of revolution is horizontl, use horizontl representtive rectngle, s shown in Figure 7 The width indictes tht is the vrile of integrtion The distnce from the center of the rectngle to the is of revolution is p, nd the height of the rectngle is h e Becuse rnges from to, the volume of the solid is d V ph d e d Appl shell method c e e 986 Integrte Tr It Eplortion A NOTE To see the dvntge of using the shell method in Emple, solve the eqution e for, ln, e e < Then use this eqution to find the volume using the disk method Comprison of Disk nd Shell Methods The disk nd shell methods cn e distinguished s follows For the disk method, the representtive rectngle is lws perpendiculr to the is of revolution, wheres for the shell method, the representtive rectngle is lws prllel to the is of revolution, s shown in Figure 7 d d V = π c (R r ) d V = π (R r ) d V = π ph d d d V = π ph d c r c R R r h p c p h Verticl is of revolution Disk method: Representtive rectngle is perpendiculr to the is of revolution Figure 7 Horizontl is of revolution Verticl is of revolution Shell method: Representtive rectngle is prllel to the is of revolution Horizontl is of revolution

25 7 CHAPTER 7 Applictions of Integrtion Often, one method is more convenient to use thn the other The following emple illustrtes cse in which the shell method is preferle EXAMPLE Shell Method Preferle For : R = r = For : R = r = () Disk method h() = + Ais of revolution () Shell method Figure 7 r Ais of revolution p() = (, ) (, ) Find the volume of the solid formed revolving the region ounded the grphs of, out the -is nd Solution In Emple in the preceding section, ou sw tht the wsher method requires two integrls to determine the volume of this solid See Figure 7() V d d Appl wsher method d d Simplif Integrte In Figure 7(), ou cn see tht the shell method requires onl one integrl to find the volume V ph d Appl shell method d,, Tr It Eplortion A Open Eplortion Integrte Suppose the region in Emple were revolved out the verticl line Would the resulting solid of revolution hve greter volume or smller volume thn the solid in Emple? Without integrting, ou should e le to reson tht the resulting solid would hve smller volume ecuse more of the revolved region would e closer to the is of revolution To confirm this, tr solving the following integrl, which gives the volume of the solid V d p FOR FURTHER INFORMATION To lern more out the disk nd shell methods, see the rticle The Disk nd Shell Method Chrles A Cle in The Americn Mthemticl Monthl MthArticle

26 SECTION 7 Volume: The Shell Method 7 Figure 7 8 ft Rottle Grph ft EXAMPLE Volume of Pontoon A pontoon is to e mde in the shpe shown in Figure 7 The pontoon is designed rotting the grph of 6, out the -is, where nd re mesured in feet Find the volume of the pontoon () Disk method () Shell method Figure 75 h() = p() = r() = R() = 6 Solution Refer to Figure 75() nd use the disk method s follows V Appl disk method 6 d Simplif 8 56 d cuic feet 5 Integrte Tr using Figure 75() to set up the integrl for the volume using the shell method Does the integrl seem more complicted? Tr It For the shell method in Emple, ou would hve to solve for in terms of in the eqution 6 Eplortion A Sometimes, solving for is ver difficult (or even impossile) In such cses ou must use verticl rectngle (of width ), thus mking the vrile of integrtion The position (horizontl or verticl) of the is of revolution then determines the method to e used This is shown in Emple 5 EXAMPLE 5 Shell Method Necessr p() = h() = + + Figure 76 (, ) Ais of revolution Find the volume of the solid formed revolving the region ounded the grphs of,, nd out the line, s shown in Figure 76 Solution In the eqution, ou cnnot esil solve for in terms of (See Section 8 on Newton s Method) Therefore, the vrile of integrtion must e, nd ou should choose verticl representtive rectngle Becuse the rectngle is prllel to the is of revolution, use the shell method nd otin V ph d d d Appl shell method Simplif Integrte Tr It Eplortion A

27 7 CHAPTER 7 Applictions of Integrtion Eercises for Section 7 The smol Click on Click on indictes n eercise in which ou re instructed to use grphing technolog or smolic computer lger sstem to view the complete solution of the eercise to print n enlrged cop of the grph In Eercises, use the shell method to set up nd evlute the integrl tht gives the volume of the solid generted revolving the plne region out the -is ,,,, 6 7, 8 9,,,,, e,,, sin, >,,,, In Eercises, use the shell method to set up nd evlute the integrl tht gives the volume of the solid generted revolving the plne region out the -is ,, 8 8,, 9 9 In Eercises, use the shell method to find the volume of the solid generted revolving the plne region out the given line,, out the line,, out the line,, out the line 5,,, out the line 6 In Eercises 5 nd 6, decide whether it is more convenient to use the disk method or the shell method to find the volume of the solid of revolution Eplin our resoning (Do not find the volume) 5 6 e In Eercises 7, use the disk or the shell method to find the volume of the solid generted revolving the region ounded the grphs of the equtions out ech given line 7 8 9,,,, 5, () the -is () the -is (c) the line,,, () the -is () the -is (c) the line,,, 5 8 () the -is () the -is (c) the line 5

28 SECTION 7 Volume: The Shell Method 7, > (hpoccloid) () the -is () the -is Writing Aout Concepts Consider solid tht is generted revolving plne region out the -is Descrie the position of representtive rectngle when using () the shell method nd () the disk method to find the volume of the solid The region in the figure is revolved out the indicted es nd line Order the volumes of the resulting solids from lest to gretest Eplin our resoning () -is () -is (c) 5 = /5 Mchine Prt A solid is generted revolving the region ounded nd out the -is A hole centered long the is of revolution, is drilled through this solid so tht one-fourth of the volume is removed Find the dimeter of the hole Mchine Prt A solid is generted revolving the region ounded 9 nd out the -is A hole centered long the is of revolution, is drilled through this solid so tht one-third of the volume is removed Find the dimeter of the hole Volume of Torus A torus is formed revolving the region ounded the circle out the line (see figure) Find the volume of this doughnut-shped solid (Hint: The integrl d represents the re of semicircle) 5 In Eercises nd, give geometric rgument tht eplins wh the integrls hve equl vlues In Eercises 5 8, () use grphing utilit to grph the plne region ounded the grphs of the equtions, nd () use the integrtion cpilities of the grphing utilit to pproimte the volume of the solid generted revolving the region out the -is 5,,, first qudrnt Think Aout It In Eercises 9 nd, determine which vlue est pproimtes the volume of the solid generted revolving the region ounded the grphs of the equtions out the -is (Mke our selection on the sis of sketch of the solid nd not performing n clcultions) 9 5 d 5 d 6 d,, 6,,, 6,, e, e,,, () () (c) (d) 75 (e) 5 tn,,, 9 () 5 () (c) 8 (d) (e) d Volume of Torus Repet Eercise for torus formed revolving the region ounded the circle r ou the line R, where r < R 5 () Use differentition to verif tht sin d sin cos C () Use the result of prt () to find the volume of the solid generted revolving ech plne region out the -is (i) 5 (ii) 6 () Use differentition to verif tht cos d cos sin C () Use the result of prt () to find the volume of the solid generted revolving ech plne region out the -is (Hint: Begin pproimting the points of intersection) (i) (ii) = cos = 5 5 = sin = cos = sin = sin = ( )

29 7 CHAPTER 7 Applictions of Integrtion In Eercises 7 5, the integrl represents the volume of solid of revolution Identif () the plne region tht is revolved nd () the is of revolution 6 7 d d 5 5 Volume of Segment of Sphere Let sphere of rdius r e cut plne, there forming segment of height h Show tht the volume of this segment is h r h 5 Volume of n Ellipsoid Consider the plne region ounded the grph of where > nd > Show tht the volume of the ellipsoid formed when this region revolves out the -is is 5 Eplortion Consider the region ounded the grphs of n, n, nd (see figure) d e d 55 Volume of Storge Shed A storge shed hs circulr se of dimeter 8 feet (see figure) Strting t the center, the interior height is mesured ever feet nd recorded in the tle Height 5 5 () Use Simpson s Rule to pproimte the volume of the shed () Note tht the roof line consists of two line segments Find the equtions of the line segments nd use integrtion to find the volume of the shed Height 5 5 Distnce from center n = n 56 Modeling Dt A pond is pproimtel circulr, with dimeter of feet (see figure) Strting t the center, the depth of the wter is mesured ever 5 feet nd recorded in the tle () Find the rtio R n of the re of the region to the re of the circumscried rectngle () Find lim R n nd compre the result with the re of the n circumscried rectngle (c) Find the volume of the solid of revolution formed revolving the region out the -is Find the rtio R n of this volume to the volume of the circumscried right circulr clinder (d) Find lim R n nd compre the result with the volume of n the circumscried clinder (e) Use the results of prts () nd (d) to mke conjecture out the shpe of the grph of n s n 5 Think Aout It Mtch ech integrl with the solid whose volume it represents, nd give the dimensions of ech solid () Right circulr cone () Torus (c) Sphere (d) Right circulr clinder (i) (iii) (v) r r r r h d r d (e) Ellipsoid (ii) (iv) R r d r h r d d Depth () Use Simpson s Rule to pproimte the volume of wter in the pond () Use the regression cpilities of grphing utilit to find qudrtic model for the depths recorded in the tle Use the grphing utilit to plot the depths nd grph the model (c) Use the integrtion cpilities of grphing utilit nd the model in prt () to pproimte the volume of wter in the pond (d) Use the result of prt (c) to pproimte the numer of gllons of wter in the pond if cuic foot of wter is pproimtel 78 gllons Depth Distnce from center

30 SECTION 7 Volume: The Shell Method Consider the grph of (see figure) Find the volumes of the solids tht re generted when the loop of this grph is revolved round () the -is, () the -is, nd (c) the line = ( ) = ( + 5) Consider the grph of 5 (see figure) Find the volume of the solid tht is generted when the loop of this grph is revolved round () the -is, () the -is, nd (c) the line 5 59 Let V nd V e the volumes of the solids tht result when the plne region ounded,, nd c c >, is revolved out the -is nd -is, respectivel Find the vlue of c for which V V Figure for 57 Figure for 58

31 76 CHAPTER 7 Applictions of Integrtion The Dutch mthemticin Christin Hugens, who invented the pendulum clock, nd Jmes Gregor (68 675), Scottish mthemticin, oth mde erl contriutions to the prolem of finding the length of rectifile curve MthBio (, ) Section 7 (, ) (, ) = = Figure 77 CHRISTIAN HUYGENS (69 695) Histor = s s = length of curve from to = n = f() ( n, n ) Arc Length nd Surfces of Revolution Find the rc length of smooth curve Find the re of surfce of revolution Arc Length In this section, definite integrls re used to find the rc lengths of curves nd the res of surfces of revolution In either cse, n rc ( segment of curve) is pproimted stright line segments whose lengths re given the fmilir Distnce Formul d A rectifile curve is one tht hs finite rc length You will see tht sufficient condition for the grph of function f to e rectifile etween, f nd, f is tht e continuous on, Such function is continuousl differentile on,, nd its grph on the intervl, is smooth curve Consider function f tht is continuousl differentile on the intervl, You cn pproimte the grph of f n line segments whose endpoints re determined the prtition f < < < < n s shown in Figure 77 B letting i i i nd i i i, ou cn pproimte the length of the grph s n i i i i i n i i i n i i i i i n i i i i This pproimtion ppers to ecome etter nd etter s n So, the length of the grph is s lim n i i i i Becuse f eists for ech in i, i, the Men Vlue Theorem gurntees the eistence of in i, i such tht f i f i fc i i i Becuse is continuous on,, it follows tht f is lso continuous (nd therefore integrle) on,, which implies tht f s lim n c i i i fc i i fc i i f d where s is clled the rc length of f etween nd

32 SECTION 7 Arc Length nd Surfces of Revolution 77 Definition of Arc Length Let the function given f represent smooth curve on the intervl, The rc length of f etween nd is s f d Similrl, for smooth curve given g, the rc length of g etween c nd d is d s g d c Becuse the definition of rc length cn e pplied to liner function, ou cn check to see tht this new definition grees with the stndrd Distnce Formul for the length of line segment This is shown in Emple Technolog (, ) (, ) EXAMPLE The Length of Line Segment Find the rc length from, to, on the grph of f m, s shown in Figure 78 Solution Becuse f() = m + The rc length of the grph of f from, to, is the sme s the stndrd Distnce Formul Figure 78 m f it follows tht s f d d Formul for rc length Integrte nd simplif which is the formul for the distnce etween two points in the plne Tr It Eplortion A TECHNOLOGY Definite integrls representing rc length often re ver difficult to evlute In this section, few emples re presented In the net chpter, with more dvnced integrtion techniques, ou will e le to tckle more difficult rc length prolems In the mentime, rememer tht ou cn lws use numericl integrtion progrm to pproimte n rc length For instnce, use the numericl integrtion feture of grphing utilit to pproimte the rc lengths in Emples nd

33 78 CHAPTER 7 Applictions of Integrtion = + 6 EXAMPLE Finding Arc Length Find the rc length of the grph of 6 on the intervl,, s shown in Figure 79 The rc length of the grph of on Figure 79 Editle Grph, FOR FURTHER INFORMATION To see how rc length cn e used to define trigonometric functions, see the rticle Trigonometr Requires Clculus, Not Vice Vers Yves Nievergelt in UMAP Modules Solution Using d d 6 ields n rc length of s d Tr It d d 6 6 Eplortion A d d 7 d Formul for rc length Simplif Integrte EXAMPLE Finding Arc Length (8, 5) 5 ( ) = (, ) The rc length of the grph of on, 8 Figure 7 Editle Grph Find the rc length of the grph of on the intervl, 8, s shown in Figure 7 Solution Begin solving for in terms of : ± Choosing the positive vlue of produces d d The -intervl, 8 corresponds to the -intervl, 5, nd the rc length is Formul for rc length d 5 d d s d c d 9 5 d 8 d Simplif Integrte Tr It Eplortion A

34 SECTION 7 Arc Length nd Surfces of Revolution 79 EXAMPLE Finding Arc Length π The rc length of the grph of on, Figure 7 Editle Grph = ln(cos ) π Find the rc length of the grph of lncos from to, s shown in Figure 7 Solution Using d d sin cos tn ields n rc length of s d Tr It d d ln sec tn ln ln 88 Eplortion A tn d sec d sec d Open Eplortion Formul for rc length Trigonometric identit Simplif Integrte EXAMPLE 5 Length of Cle 5 Ctenr: = 5 cosh 5 An electric cle is hung etween two towers tht re feet prt, s shown in Figure 7 The cle tkes the shpe of ctenr whose eqution is 75e 5 e 5 5 cosh 5 Find the rc length of the cle etween the two towers e5 e 5, Solution Becuse ou cn write Figure 7 nd e75 e 75 e75 e 75 e5 e 5 Therefore, the rc length of the cle is s d e 5 e 5 d 75 e 5 e 5 Formul for rc length Integrte 5e e 5 feet Tr It Eplortion A

35 8 CHAPTER 7 Applictions of Integrtion Are of Surfce of Revolution In Sections 7 nd 7, integrtion ws used to clculte the volume of solid of revolution You will now look t procedure for finding the re of surfce of revolution Definition of Surfce of Revolution If the grph of continuous function is revolved out line, the resulting surfce is surfce of revolution Figure 7 L r r Rottle Grph Ais of revolution The re of surfce of revolution is derived from the formul for the lterl surfce re of the frustum of right circulr cone Consider the line segment in Figure 7, where L is the length of the line segment, r is the rdius t the left end of the line segment, nd r is the rdius t the right end of the line segment When the line segment is revolved out its is of revolution, it forms frustum of right circulr cone, with where S r L r r r Lterl surfce re of frustum Averge rdius of frustum (In Eercise 6, ou re sked to verif the formul for S ) Suppose the grph of function f, hving continuous derivtive on the intervl,, is revolved out the -is to form surfce of revolution, s shown in Figure 7 Let e prtition of,, with suintervls of width i Then the line segment of length L i i i genertes frustum of cone Let r i e the verge rdius of this frustum B the Intermedite Vlue Theorem, point d i eists (in the ith suintervl) such tht r i f d i The lterl surfce re S i of the frustum is S i r i L i f d i i i f d i i i i = f() L i i i = i i = n Ais of Figure 7 revolution Rottle Grph

36 SECTION 7 Arc Length nd Surfces of Revolution 8 Ais of revolution Ais of revolution Figure 75 r = f() r = = f() (, f()) = f() (, f()) B the Men Vlue Theorem, point eists in i, i such tht fc i f i f i i i i i So, S i f d i fc i i, nd the totl surfce re cn e pproimted S n i f d i fc i i It cn e shown tht the limit of the right side s n is S f f d In similr mnner, if the grph of f is revolved out the -is, then S is S f d c i In oth formuls for S, ou cn regrd the products f nd s the circumference of the circle trced point, on the grph of f s it is revolved out the - or -is (Figure 75) In one cse the rdius is r f, nd in the other cse the rdius is r Moreover, ppropritel djusting r, ou cn generlize the formul for surfce re to cover n horizontl or verticl is of revolution, s indicted in the following definition Definition of the Are of Surfce of Revolution Let f hve continuous derivtive on the intervl, The re S of the surfce of revolution formed revolving the grph of f out horizontl or verticl is is S r f d is function of where r is the distnce etween the grph of f nd the is of revolution If g on the intervl c, d, then the surfce re is d S r g d is function of c where r is the distnce etween the grph of g nd the is of revolution The formuls in this definition re sometimes written s S r ds is function of nd d S r) ds c is function of where ds f d nd ds g d, respectivel

37 8 CHAPTER 7 Applictions of Integrtion EXAMPLE 6 The Are of Surfce of Revolution Find the re of the surfce formed revolving the grph of f on the intervl, out the -is, s shown in Figure 76 f() = (, ) Solution The distnce etween the -is nd the grph of f is r f, nd ecuse f, the surfce re is Figure 76 r() = f() Ais of revolution S d d r f d Formul for surfce re Simplif Integrte Rottle Grph Tr It EXAMPLE 7 Eplortion A The Are of Surfce of Revolution r() = Ais of revolution Figure 77 (, ) f() = Find the re of the surfce formed revolving the grph of f on the intervl, out the -is, s shown in Figure 77 Solution In this cse, the distnce etween the grph of f nd the -is is r Using f, ou cn determine tht the surfce re is S r f d d 8 d Formul for surfce re Simplif Integrte Rottle Grph Tr It Eplortion A

38 SECTION 7 Arc Length nd Surfces of Revolution 8 Eercises for Section 7 The smol Click on Click on indictes n eercise in which ou re instructed to use grphing technolog or smolic computer lger sstem to view the complete solution of the eercise to print n enlrged cop of the grph In Eercises nd, find the distnce etween the points using () the Distnce Formul nd () integrtion,, 5,,, In Eercises, find the rc length of the grph of the function over the indicted intervl ,, 8,, 7 9 lnsin,, lncos,, In Eercises 5, () grph the function, highlighting the prt indicted the given intervl, () find definite integrl tht represents the rc length of the curve over the indicted intervl nd oserve tht the integrl cnnot e evluted with the techniques studied so fr, nd (c) use the integrtion cpilities of grphing utilit to pproimte the rc length e e, ln e e,,,, = / + = / 6 8,,, ln, ln , 8 = / = Approimtion In Eercises 5 nd 6, determine which vlue est pproimtes the length of the rc represented the integrl (Mke our selection on the sis of sketch of the rc nd not performing n clcultions) 5 d 5 d d () 5 () 5 (c) (d) (e) 6, sin, e, ln, rctn, 6, d d tn d () () (c) (d) (e) Approimtion In Eercises 7 nd 8, pproimte the rc length of the grph of the function over the intervl [, ] in four ws () Use the Distnce Formul to find the distnce etween the endpoints of the rc () Use the Distnce Formul to find the lengths of the four line segments connecting the points on the rc when,,,, nd Find the sum of the four lengths (c) Use Simpson s Rule with n to pproimte the integrl ielding the indicted rc length (d) Use the integrtion cpilities of grphing utilit to pproimte the integrl ielding the indicted rc length 7 f 8 f 9 () Use grphing utilit to grph the function f () Cn ou integrte with respect to to find the rc length of the grph of f on the intervl, 8? Eplin (c) Find the rc length of the grph of f on the intervl, 8 Astroid Find the totl length of the grph of the stroid cos, / + / =

39 8 CHAPTER 7 Applictions of Integrtion Think Aout It The figure shows the grphs of the functions nd,,, 8 5 on the intervl, To print n enlrged cop of the grph, select the MthGrph utton () Lel the functions () List the functions in order of incresing rc length (c) Verif our nswer in prt () pproimting ech rc length ccurte to three deciml plces Think Aout It Eplin wh the two integrls re equl e d e d Use the integrtion cpilities of grphing utilit to verif tht the integrls re equl Length of Pursuit A fleeing oject leves the origin nd moves up the -is (see figure) At the sme time, pursuer leves the point (, ) nd lws moves towrd the fleeing oject The pursuer s speed is twice tht of the fleeing oject The eqution of the pth is modeled How fr hs the fleeing oject trveled when it is cught? Show tht the pursuer hs trveled twice s fr ft Figure for 5 Figure for 6 6 Length of Gtew Arch The Gtew Arch in St Louis Missouri, is modeled (See Section 58, Section Project: St Louis Arch) Find the length of this curve (see figure) 7 Find the rc length from, clockwise to, 5 long the circle 9 8 Find the rc length from, clockwise to, long the circle 5 Show tht the result is one-fourth the circumference of the circle In Eercises 9, set up nd evlute the definite integrl for the re of the surfce generted revolving the curve out the -is 9 = cosh, (99, ) (99, ) 6 6 = (, 65) 6 8 = (/ / + ) = (e / + e / ) Rottle Grph 6, Rottle Grph, 6 Figure for Figure for Rottle Grph In Eercises nd, set up nd evlute the definite integr for the re of the surfce generted revolving the curve out the -is Roof Are A rn is feet long nd feet wide (see figure) A cross section of the roof is the inverted ctenr e e Find the numer of squre feet of roofing on the rn 5 Length of Ctenr Electricl wires suspended etween two towers form ctenr (see figure) modeled the eqution cosh, where nd re mesured in meters The towers re meters prt Find the length of the suspended cle 8 6 Rottle Grph = = 9 Rottle Grph

40 SECTION 7 Arc Length nd Surfces of Revolution 85 In Eercises 5 nd 6, use the integrtion cpilities of grphing utilit to pproimte the surfce re of the solid of revolution = / / 5 6 Function sin revolved out the -is ln revolved out the -is,, e Writing Aout Concepts 7 Define rectifile curve Intervl 8 Wht preclculus formul nd representtive element re used to develop the integrtion formul for rc length? 9 Wht preclculus formul nd representtive element re used to develop the integrtion formul for the re of surfce of revolution? 5 The grphs of the functions f nd f on the intervl, ] re shown in the figure The grph of ech is revolved out the -is Which surfce of revolution hs the greter surfce re? Eplin f Figure for 55 Rottle Grph 56 Think Aout It Consider the eqution 9 () Use grphing utilit to grph the eqution () Set up the definite integrl for finding the first qudrnt rc length of the grph in prt () (c) Compre the intervl of integrtion in prt () nd the domin of the integrnd Is it possile to evlute the definite integrl? Is it possile to use Simpson s Rule to evlute the definite integrl? Eplin (You will lern how to evlute this tpe of integrl in Section 88) 57 Modeling Dt The circumference C (in inches) of vse is mesured t three-inch intervls strting t its se The mesurements re shown in the tle, where is the vertic distnce in inches from the se f C A right circulr cone is generted revolving the region ounded hr, h, nd out the -is Verif tht the lterl surfce re of the cone is S rr h 5 A sphere of rdius r is generted revolving the grph of r out the -is Verif tht the surfce re of the sphere is r 5 Find the re of the zone of sphere formed revolving the grph of 9,, out the -is 5 Find the re of the zone of sphere formed revolving the grph of r,, out the -is Assume tht < r 55 Bul Design An ornmentl light ul is designed revolving the grph of, out the -is, where nd re mesured in feet (see figure) Find the surfce re of the ul nd use the result to pproimte the mount of glss needed to mke the ul (Assume tht the glss is 5 inch thick) () Use the dt to pproimte the volume of the vse summing the volumes of pproimting disks () Use the dt to pproimte the outside surfce re (ecluding the se) of the vse summing the outside surfce res of pproimting frustums of right circulr cones (c) Use the regression cpilities of grphing utilit to find cuic model for the points, r where r C Use the grphing utilit to plot the points nd grph the model (d) Use the model in prt (c) nd the integrtion cpilities of grphing utilit to pproimte the volume nd outside surfce re of the vse Compre the results with our nswers in prts () nd () 58 Modeling Dt Propert ounded two perpendiculr rods nd strem is shown in the figure on the net pge Al distnces re mesured in feet () Use the regression cpilities of grphing utilit to fit fourth-degree polnomil to the pth of the strem () Use the model in prt () to pproimte the re of the propert in cres (c) Use the integrtion cpilities of grphing utilit to find the length of the strem tht ounds the propert

41 86 CHAPTER 7 Applictions of Integrtion 6 (, 5) (5, ) (5, 9) (,5) (5, 6) (, 9) (, 75) Figure for Let R e the region ounded, the -is,, nd, where > Let D e the solid formed when R is revolved out the -is () Find the volume V of D () Write the surfce re S s n integrl (c) Show tht V pproches finite limit s (d) Show tht S s 6 () Given circulr sector with rdius L nd centrl ngle (see figure), show tht the re of the sector is given S L () B joining the stright line edges of the sector in prt (), right circulr cone is formed (see figure) nd the lterl surfce re of the cone is the sme s the re of the sector Show tht the re is S rl, where r is the rdius of the se of the cone (Hint: The rc length of the sector equls the circumference of the se of the cone) L (5, 5) (, ) 6 r L 6 Individul Project Select solid of revolution from everd life Mesure the rdius of the solid t minimum of seven points long its is Use the dt to pproimte the volume of the solid nd the surfce re of the lterl sides of the solid 6 Writing Red the rticle Arc Length, Are nd the Arcsine Function Andrew M Rockett in Mthemtics Mgzine Then write prgrph eplining how the rcsine function cn e defined in terms of n rc length MthArticle 6 Astroid Find the re of the surfce formed revolving the portion in the first qudrnt of the grph of 8 out the -is Figure for 6 Figure for 6 Rottle Grph 6 Consider the grph of (see figure) Find the re of the surfce formed when the loop of this grph is revolved round the -is 65 Suspension Bridge A cle for suspension ridge hs the shpe of prol with eqution k Let h represent the height of the cle from its lowest point to its highest point nd let w represent the totl spn of the ridge (see figure) Show tht the length C of the cle is given w h C w d 5 6 = ( ) Figure for 6() Figure for 6() (c) Use the result of prt () to verif tht the formul for the lterl surfce re of the frustum of cone with slnt height L nd rdii r nd r (see figure) is S r r L (Note: This formul ws used to develop the integrl for finding the surfce re of surfce of revolution) L r r Rottle Grph w 66 Suspension Bridge The Humer Bridge, locted in the United Kingdom nd opened in 98, hs min spn of ou meters Ech of its towers hs height of out 55 meters Use these dimensions, the integrl in Eercise 65, nd the integrtion cpilities of grphing utilit to pproimte the length of prolic cle long the min spn h Rottle Grph Ais of revolution Putnm Em Chllenge 67 Find the length of the curve from the origin to the poin where the tngent mkes n ngle of 5 with the -is This prolem ws composed the Committee on the Putnm Prize Competition The Mthemticl Assocition of Americ All rights reserved

42 SECTION 75 Work 87 Section 75 Work Find the work done constnt force Find the work done vrile force Work Done Constnt Force The concept of work is importnt to scientists nd engineers for determining the energ needed to perform vrious jos For instnce, it is useful to know the mount of work done when crne lifts steel girder, when spring is compressed, when rocket is propelled into the ir, or when truck pulls lod long highw In generl, work is done force when it moves n oject If the force pplied to the oject is constnt, then the definition of work is s follows Definition of Work Done Constnt Force If n oject is moved distnce D in the direction of n pplied constnt force F, then the work W done the force is defined s W FD There re mn tpes of forces centrifugl, electromotive, nd grvittionl, to nme few A force cn e thought of s push or pull; force chnges the stte of rest or stte of motion of od For grvittionl forces on Erth, it is common to use units of mesure corresponding to the weight of n oject EXAMPLE Lifting n Oject Determine the work done in lifting 5-pound oject feet 5 l 5 l ft The work done in lifting 5-pound oject feet is foot-pounds Figure 78 Simultion Solution The mgnitude of the required force F is the weight of the oject, s shown in Figure 78 So, the work done in lifting the oject feet is W FD 5 foot-pounds Work forcedistnce Force 5 pounds, distnce feet Tr It Eplortion A Eplortion B In the US mesurement sstem, work is tpicll epressed in foot-pounds (ft-l), inch-pounds, or foot-tons In the centimeter-grm-second (C-G-S) sstem, the sic unit of force is the dne the force required to produce n ccelertion of centimeter per second per second on mss of grm In this sstem, work is tpicll epressed in dne-centimeters (ergs) or newton-meters (joules), where joule 7 ergs EXPLORATION How Much Work? In Emple, foot-pounds of work ws needed to lift the 5-pound oject feet verticll off the ground Suppose tht once ou lifted the oject, ou held it nd wlked horizontl distnce of feet Would this require n dditionl foot-pounds of work? Eplin our resoning

43 88 CHAPTER 7 Applictions of Integrtion F() The mount of force chnges s n oject chnges position Figure 79 Work Done Vrile Force In Emple, the force involved ws constnt If vrile force is pplied to n oject, clculus is needed to determine the work done, ecuse the mount of force chnges s the oject chnges position For instnce, the force required to compress spring increses s the spring is compressed Suppose tht n oject is moved long stright line from to continuousl vring force F Let e prtition tht divides the intervl, into n suintervls determined < < < < n nd let i i i For ech i, choose c i such tht i c i i Then t c i the force is given Fc i Becuse F is continuous, ou cn pproimte the work done in moving the oject through the ith suintervl the increment W i Fc i i s shown in Figure 79 So, the totl work done s the oject moves from to is pproimted W n W i i n Fc i i i This pproimtion ppers to ecome etter nd etter s n So, the work done is W lim n i F d Fc i i EMILIE DE BRETEUIL (76 79) Another mjor work de Breteuil ws the trnsltion of Newton s Philosophie Nturlis Principi Mthemtic into French Her trnsltion nd commentr gretl contriuted to the cceptnce of Newtonin science in Europe MthBio Definition of Work Done Vrile Force If n oject is moved long stright line continuousl vring force F, then the work W done the force s the oject is moved from to is W lim n i F d W i The remining emples in this section use some well-known phsicl lws The discoveries of mn of these lws occurred during the sme period in which clculus ws eing developed In fct, during the seventeenth nd eighteenth centuries, there ws little difference etween phsicists nd mthemticins One such phsicistmthemticin ws Emilie de Breteuil Breteuil ws instrumentl in snthesizing the work of mn other scientists, including Newton, Leiniz, Hugens, Kepler, nd Descrtes Her phsics tet Institutions ws widel used for mn ers

44 SECTION 75 Work 89 The following three lws of phsics were developed Roert Hooke (65 7), Isc Newton (6 77), nd Chrles Coulom (76 86) Hooke s Lw: The force F required to compress or stretch spring (within its elstic limits) is proportionl to the distnce d tht the spring is compressed or stretched from its originl length Tht is, F kd where the constnt of proportionlit k specific nture of the spring (the spring constnt) depends on the Newton s Lw of Universl Grvittion: The force F of ttrction etween two prticles of msses m nd m is proportionl to the product of the msses nd inversel proportionl to the squre of the distnce d etween the two prticles Tht is, F k m m d EXPLORATION The work done in compressing the spring in Emple from inches to 6 inches is 75 inch-pounds Should the work done in compressing the spring from inches to inches e more thn, the sme s, or less thn this? Eplin If m nd m re given in grms nd d in centimeters, F will e in dnes for vlue of k cuic centimeter per grm-second squred Coulom s Lw: The force etween two chrges q nd q in vcuum is proportionl to the product of the chrges nd inversel proportionl to the squre of the distnce d etween the two chrges Tht is, F k q q d If q nd q re given in electrosttic units nd d in centimeters, F will e in dnes for vlue of k EXAMPLE Compressing Spring Nturl length (F = ) 5 Compressed inches (F = 75) 5 Compressed inches (F = 5) Figure 75 5 A force of 75 pounds compresses spring inches from its nturl length of 5 inches Find the work done in compressing the spring n dditionl inches Solution B Hooke s Lw, the force F required to compress the spring units (from its nturl length) is F k Using the given dt, it follows tht F 75 k nd so k 5 nd F 5, s shown in Figure 75 To find the increment of work, ssume tht the force required to compress the spring over smll increment is nerl constnt So, the increment of work is W forcedistnce increment 5 Becuse the spring is compressed from to 6 inches less thn its nturl length, the work required is 6 F d 5 d Formul for work W inch-pounds Note tht ou do not integrte from to 6 ecuse ou were sked to determine the work done in compressing the spring n dditionl inches (not including the first inches) Tr It Eplortion A Eplortion B Open Eplortion

45 9 CHAPTER 7 Applictions of Integrtion EXAMPLE Moving Spce Module into Orit mi 8 mi A spce module weighs 5 metric tons on the surfce of Erth How much work is done in propelling the module to height of 8 miles ove Erth, s shown in Figure 75? (Use miles s the rdius of Erth Do not consider the effect of ir resistnce or the weight of the propellnt) Not drwn to scle 8 Figure 75 Solution Becuse the weight of od vries inversel s the squre of its distnce from the center of Erth, the force F eerted grvit is F C C is the constnt of proportionlit Becuse the module weighs 5 metric tons on the surfce of Erth nd the rdius of Erth is pproimtel miles, ou hve So, the increment of work is Finll, ecuse the module is propelled from to 8 miles, the totl work done is 8,, F d d Formul for work W 5 C,, C W forcedistnce increment,,,, 8 5, 6,, mile-tons 6 foot-pounds Integrte In the C-G-S sstem, using conversion fctor of foot-pound 558 joules, the work done is W 578 joules Tr It Eplortion A Eplortion B The solutions to Emples nd conform to our development of work s the summtion of increments in the form W forcedistnce increment F Another w to formulte the increment of work is W force incrementdistnce F This second interprettion of W is useful in prolems involving the movement of nonrigid sustnces such s fluids nd chins

46 SECTION 75 Work 9 EXAMPLE Empting Tnk of Oil A sphericl tnk of rdius 8 feet is hlf full of oil tht weighs 5 pounds per cuic foot Find the work required to pump oil out through hole in the top of the tnk 8 6 Solution Consider the oil to e sudivided into disks of thickness nd rdius, s shown in Figure 75 Becuse the increment of force for ech disk is given its weight, ou hve F weight 6 5 pounds cuic foot volume 5 pounds For circle of rdius 8 nd center t, 8, ou hve 8 Figure 75 Rottle Grph nd ou cn write the force increment s F In Figure 75, note tht disk feet from the ottom of the tnk must e moved distnce of 6 feet So, the increment of work is W F Becuse the tnk is hlf full, rnges from to 8, nd the work required to empt the tnk is 8 W 556 d 5 8 5,6 589,78 foot-pounds 8 Tr It Eplortion A To estimte the resonleness of the result in Emple, consider tht the weight of the oil in the tnk is volumedensit 8 5 5,665 pounds Lifting the entire hlf-tnk of oil 8 feet would involve work of 85,665 8,9 foot-pounds Becuse the oil is ctull lifted etween 8 nd 6 feet, it seems resonle tht the work done is 589,78 foot-pounds

47 9 CHAPTER 7 Applictions of Integrtion EXAMPLE 5 Lifting Chin A -foot chin weighing 5 pounds per foot is ling coiled on the ground How much work is required to rise one end of the chin to height of feet so tht it is full etended, s shown in Figure 75? Work required to rise one end of the chin Figure 75 Solution Imgine tht the chin is divided into smll sections, ech of length Then the weight of ech section is the increment of force F weight 5 pounds foot length 5 Becuse tpicl section (initill on the ground) is rised to height of, the increment of work is W force incrementdistnce 5 5 Becuse rnges from to, the totl work is W 5 d 5 Tr It 5 Eplortion A foot-pounds Work done epnding gs Figure 75 r Gs In the net emple ou will consider piston of rdius r in clindricl csing, s shown in Figure 75 As the gs in the clinder epnds, the piston moves nd work is done If p represents the pressure of the gs (in pounds per squre foot) ginst the piston hed nd V represents the volume of the gs (in cuic feet), the work increment involved in moving the piston feet is W forcedistnce increment F pr p V So, s the volume of the gs epnds from V to V, the work done in moving the piston is W V V Assuming the pressure of the gs to e inversel proportionl to its volume, ou hve p kv nd the integrl for work ecomes W V V p dv k V dv EXAMPLE 6 Work Done n Epnding Gs A quntit of gs with n initil volume of cuic foot nd pressure of 5 pounds per squre foot epnds to volume of cuic feet Find the work done the gs (Assume tht the pressure is inversel proportionl to the volume) Solution Becuse p kv nd p 5 when V, ou hve k 5 So, the work is W V V k V dv 5 V dv 5 ln V 66 foot-pounds Tr It Eplortion A

48 SECTION 75 Work 9 Eercises for Section 75 The smol Click on Click on indictes n eercise in which ou re instructed to use grphing technolog or smolic computer lger sstem to view the complete solution of the eercise to print n enlrged cop of the grph Constnt Force In Eercises, determine the work done the constnt force A -pound g of sugr is lifted feet An electric hoist lifts 8-pound cr feet A force of newtons is required to slide cement lock meters in construction project The locomotive of freight trin pulls its crs with constnt force of 9 tons distnce of one-hlf mile Writing Aout Concepts 5 Stte the definition of work done constnt force 6 Stte the definition of work done vrile force 7 The grphs show the force (in pounds) required to move n oject 9 feet long the -is Order the force functions from the one tht ields the lest work to the one tht ields the most work without doing n clcultions Eplin our resoning () (c) 8 6 F F F 6 8 F = F i 8 Verif our nswer to Eercise 7 clculting the work for ech force function Hooke s Lw In Eercises 9 6, use Hooke s Lw to determine the vrile force in the spring prolem 9 A force of 5 pounds compresses 5-inch spring totl of inches How much work is done in compressing the spring 7 inches? How much work is done in compressing the spring in Eercise 9 from length of inches to length of 6 inches? A force of 5 newtons stretches spring centimeters How much work is done in stretching the spring from centimeters to 5 centimeters? () (d) 6 8 F F F = F A force of 8 newtons stretches spring 7 centimeters on mechnicl device for driving fence posts Find the work done in stretching the spring the required 7 centimeters A force of pounds stretches spring 9 inches in n eercise mchine Find the work done in stretching the spring foo from its nturl position An overhed grge door hs two springs, one on ech side of the door A force of 5 pounds is required to stretch ech spring foot Becuse of the pulle sstem, the springs stretch onl one-hlf the distnce the door trvels The door moves totl of 8 feet nd the springs re t their nturl length when the door is open Find the work done the pir of springs 5 Eighteen foot-pounds of work is required to stretch spring inches from its nturl length Find the work required to stretch the spring n dditionl inches 6 Seven nd one-hlf foot-pounds of work is required to compress spring inches from its nturl length Find the work required to compress the spring n dditionl one-hlf inch 7 Propulsion Neglecting ir resistnce nd the weight of the propellnt, determine the work done in propelling five-ton stellite to height of () miles ove Erth () miles ove Erth 8 Propulsion Use the informtion in Eercise 7 to write the work W of the propulsion sstem s function of the height h of the stellite ove Erth Find the limit (if it eists) of W s h pproches infinit 9 Propulsion Neglecting ir resistnce nd the weight of the propellnt, determine the work done in propelling -ton stellite to height of (), miles ove Erth (), miles ove Erth Propulsion A lunr module weighs tons on the surfce of Erth How much work is done in propelling the module from the surfce of the moon to height of 5 miles? Consider the rdius of the moon to e miles nd its force of grvit to e one-sith tht of Erth Pumping Wter A rectngulr tnk with se feet 5 feet nd height of feet is full of wter (see figure) The wter weighs 6 pounds per cuic foot How much work is done in pumping wter out over the top edge in order to empt () hlf of the tnk? () ll of the tnk? 5 ft ft ft

49 9 CHAPTER 7 Applictions of Integrtion Think Aout It Eplin wh the nswer in prt () of Eercise is not twice the nswer in prt () Pumping Wter A clindricl wter tnk meters high with rdius of meters is uried so tht the top of the tnk is meter elow ground level (see figure) How much work is done in pumping full tnk of wter up to ground level? (The wter weighs 98 newtons per cuic meter) Figure for Figure for Pumping Wter Suppose the tnk in Eercise is locted on tower so tht the ottom of the tnk is meters ove the level of strem (see figure) How much work is done in filling the tnk hlf full of wter through hole in the ottom, using wter from the strem? 5 Pumping Wter An open tnk hs the shpe of right circulr cone (see figure) The tnk is 8 feet cross the top nd 6 feet high How much work is done in empting the tnk pumping the wter over the top edge? 5 Rottle Grph 6 Figure for 5 Figure for 8 6 Pumping Wter Wter is pumped in through the ottom of the tnk in Eercise 5 How much work is done to fill the tnk () to depth of feet? Rottle Grph Ground level 5 6 () from depth of feet to depth of 6 feet? 7 Pumping Wter A hemisphericl tnk of rdius 6 feet is positioned so tht its se is circulr How much work is required to fill the tnk with wter through hole in the se if the wter source is t the se? 8 Pumping Diesel Fuel The fuel tnk on truck hs trpezoidl cross sections with dimensions (in feet) shown in the figure Assume tht n engine is pproimtel feet ove the top of the fuel tnk nd tht diesel fuel weighs pproimtel 5 pounds per cuic foot Find the work done the fuel pump in rising full tnk of fuel to the level of the engine m Rottle Grph Rottle Grph Pumping Gsoline In Eercises 9 nd, find the work done in pumping gsoline tht weighs pounds per cuic foot (Hint: Evlute one integrl geometric formul nd the other oserving tht the integrnd is n odd function) 9 A clindricl gsoline tnk feet in dimeter nd feet long is crried on the ck of truck nd is used to fuel trctors The is of the tnk is horizontl The opening on the trctor tnk is 5 feet ove the top of the tnk in the truck Find the work done in pumping the entire contents of the fuel tnk into trctor The top of clindricl storge tnk for gsoline t service sttion is feet elow ground level The is of the tnk is horizontl nd its dimeter nd length re 5 feet nd feet respectivel Find the work done in pumping the entire contents of the full tnk to height of feet ove ground level Lifting Chin In Eercises, consider 5-foot chin tht weighs pounds per foot hnging from winch 5 feet ove ground level Find the work done the winch in winding up the specified mount of chin Wind up the entire chin Wind up one-third of the chin Run the winch until the ottom of the chin is t the -foo level Wind up the entire chin with 5-pound lod ttched to it Lifting Chin In Eercises 5 nd 6, consider 5-foot hnging chin tht weighs pounds per foot Find the work done in lifting the chin verticll to the indicted position 5 Tke the ottom of the chin nd rise it to the 5-foot level leving the chin douled nd still hnging verticll (see figure) Repet Eercise 5 rising the ottom of the chin to the -foot level Demolition Crne In Eercises 7 nd 8, consider demolition crne with 5-pound ll suspended from -foot cle tht weighs pound per foot 7 Find the work required to wind up 5 feet of the pprtus 8 Find the work required to wind up ll feet of the pprtus 5

50 SECTION 75 Work 95 Bole s Lw In Eercises 9 nd, find the work done the gs for the given volume nd pressure Assume tht the pressure is inversel proportionl to the volume (See Emple 6) 9 A quntit of gs with n initil volume of cuic feet nd pressure of pounds per squre foot epnds to volume of cuic feet A quntit of gs with n initil volume of cuic foot nd pressure of 5 pounds per squre foot epnds to volume of cuic feet Electric Force Two electrons repel ech other with force tht vries inversel s the squre of the distnce etween them One electron is fied t the point, Find the work done in moving the second electron from, to, Modeling Dt The hdrulic clinder on woodsplitter hs four-inch ore (dimeter) nd stroke of feet The hdrulic pump cretes mimum pressure of pounds per squre inch Therefore, the mimum force creted the clinder is 8 pounds () Find the work done through one etension of the clinder given tht the mimum force is required () The force eerted in splitting piece of wood is vrile Mesurements of the force otined when piece of wood ws split re shown in the tle The vrile mesures the etension of the clinder in feet, nd F is the force in pounds Use Simpson s Rule to pproimte the work done in splitting the piece of wood Tle for () (c) Use the regression cpilities of grphing utilit to find fourth-degree polnomil model for the dt Plot the dt nd grph the model (d) Use the model in prt (c) to pproimte the etension o the clinder when the force is mimum (e) Use the model in prt (c) to pproimte the work done in splitting the piece of wood Hdrulic Press In Eercises 6, use the integrtion cpilities of grphing utilit to pproimte the work done press in mnufcturing process A model for the vrile force F (in pounds) nd the distnce (in feet) the press moves is given 5 F Force,, 5,, 5 F 8 ln F e F 5 Intervl F sinh 5

51 96 CHAPTER 7 Applictions of Integrtion Section 76 Moments, Centers of Mss, nd Centroids Understnd the definition of mss Find the center of mss in one-dimensionl sstem Find the center of mss in two-dimensionl sstem Find the center of mss of plnr lmin Use the Theorem of Pppus to find the volume of solid of revolution Mss In this section ou will stud severl importnt pplictions of integrtion tht re relted to mss Mss is mesure of od s resistnce to chnges in motion, nd is independent of the prticulr grvittionl sstem in which the od is locted However, ecuse so mn pplictions involving mss occur on Erth s surfce, n oject s mss is sometimes equted with its weight This is not technicll correct Weight is tpe of force nd s such is dependent on grvit Force nd mss re relted the eqution Force mssccelertion The tle elow lists some commonl used mesures of mss nd force, together with their conversion fctors Sstem of Mesure of Mesurement Mss Mesure of Force US Interntionl C-G-S Slug Kilogrm Grm Pound slugftsec Newton kilogrmmsec Dne grmcmsec Conversions: pound 8 newtons slug 59 kilogrms newton 8 pound kilogrm 685 slug dne 8 pound grm 685 slug dne newton foot 8 meter EXAMPLE Mss on the Surfce of Erth Find the mss (in slugs) of n oject whose weight t se level is pound Solution produces Mss Using feet per second per second s the ccelertion due to grvit force ccelertion pound feet per second per second 5 5 slug Force mssccelertion pound foot per second per second Becuse mn pplictions involving mss occur on Erth s surfce, this mount of mss is clled pound mss Tr It Eplortion A

52 SECTION 76 Moments, Centers of Mss, nd Centroids 97 kg kg m P m The seesw will lnce when the left nd the right moments re equl Figure 755 Center of Mss in One-Dimensionl Sstem You will now consider two tpes of moments of mss the moment out point nd the moment out line To define these two moments, consider n idelized sitution in which mss m is concentrted t point If is the distnce etween this point mss nd nother point P, the moment of m out the point P is Moment m nd is the length of the moment rm The concept of moment cn e demonstrted simpl seesw, s shown in Figure 755 A child of mss kilogrms sits meters to the left of fulcrum P, nd n older child of mss kilogrms sits meters to the right of P From eperience, ou know tht the seesw will egin to rotte clockwise, moving the lrger child down This rottion occurs ecuse the moment produced the child on the left is less thn the moment produced the child on the right Left moment Right moment 6 kilogrm-meters kilogrm-meters To lnce the seesw, the two moments must e equl For emple, if the lrger child moved to position meters from the fulcrum, the seesw would lnce, ecuse ech child would produce moment of kilogrm-meters To generlize this, ou cn introduce coordinte line on which the origin corresponds to the fulcrum, s shown in Figure 756 Suppose severl point msses re locted on the -is The mesure of the tendenc of this sstem to rotte out the origin is the moment out the origin, nd it is defined s the sum of the n products m i i M m m m n n m m m m n n m n n If m m m n n, the sstem is in equilirium Figure 756 If M is, the sstem is sid to e in equilirium The concept of equilirium is demonstrted in the simultion elow Simultion For sstem tht is not in equilirium, the center of mss is defined s the point t which the fulcrum could e relocted to ttin equilirium If the sstem were trnslted units, ech coordinte i would ecome i, nd ecuse the moment of the trnslted sstem is, ou hve n m i i n m i i n m i i i i Solving for produces m i n i i n m i i moment of sstem out origin totl mss of sstem If m m m n n, the sstem is in equilirium

53 98 CHAPTER 7 Applictions of Integrtion Moments nd Center of Mss: One-Dimensionl Sstem Let the point msses m, m,, m n e locted t,,, n The moment out the origin is M m m m n n The center of mss is M where m m m m n is the m, totl mss of the sstem EXAMPLE The Center of Mss of Liner Sstem Find the center of mss of the liner sstem shown in Figure 757 m m m m Figure Solution The moment out the origin is M m m m m Becuse the totl mss of the sstem is m 5 5, the center of mss is M m Tr It Eplortion A NOTE In Emple, where should ou locte the fulcrum so tht the point msses will e in equilirium? Rther thn define the moment of mss, ou could define the moment of force In this contet, the center of mss is clled the center of grvit Suppose tht sstem of point msses m, m,, m n is locted t,,, n Then, ecuse force mssccelertion, the totl force of the sstem is F m m m n m The torque (moment) out the origin is T m m m n n M nd the center of grvit is T F M m M m So, the center of grvit nd the center of mss hve the sme loction

54 SECTION 76 Moments, Centers of Mss, nd Centroids 99 Center of Mss in Two-Dimensionl Sstem (, ) You cn etend the concept of moment to two dimensions considering sstem of msses locted in the -plne t the points,,,,, n, n, s shown in Figure 758 Rther thn defining single moment (with respect to the origin), two moments re defined one with respect to the -is nd one with respect to the -is m m n m ( n, n ) (, ) In two-dimensionl sstem, there is moment out the -is, M, nd moment out the -is, M Figure 758 Moments nd Center of Mss: Two-Dimensionl Sstem Let the point msses m, m,, m n e locted t,,,,, n, n The moment out the -is is M m m m n n The moment out the -is is M m m m n n The center of mss, (or center of grvit) is M m nd M m where m m m m n is the totl mss of the sstem The moment of sstem of msses in the plne cn e tken out n horizontl or verticl line In generl, the moment out line is the sum of the product of the msses nd the directed distnces from the points to the line Moment m m m n n Moment m m m n n Horizontl line Verticl line EXAMPLE The Center of Mss of Two-Dimensionl Sstem m = m = 9 ( 5, ) (, ) m = (, ) 5 m = 6 (, ) Figure 759 Find the center of mss of sstem of point msses m 6, m, m, nd m 9, locted t,,,, 5,, nd, s shown in Figure 759 Solution So, nd m 6 9 M M 6 ( 9 M m M m 5 5 nd so the center of mss is 5, 5 Mss Moment out -is Moment out -is Tr It Eplortion A

55 5 CHAPTER 7 Applictions of Integrtion (, ) You cn think of the center of mss, of lmin s its lncing point For circulr lmin, the center of mss is the center of the circle For rectngulr lmin, the center of mss is the center of the rectngle Figure 76 i ( i, f( i )) (, ) ( i, i ) ( i, g( i )) i Plnr lmin of uniform densit Figure 76 f g Center of Mss of Plnr Lmin So fr in this section ou hve ssumed the totl mss of sstem to e distriuted t discrete points in plne or on line Now consider thin, flt plte of mteril of constnt densit clled plnr lmin (see Figure 76) Densit is mesure of mss per unit of volume, such s grms per cuic centimeter For plnr lmins, however, densit is considered to e mesure of mss per unit of re Densit is denoted, the lowercse Greek letter rho Consider n irregulrl shped plnr lmin of uniform densit, ounded the grphs of f, g, nd, s shown in Figure 76 The mss of this region is given where A is the re of the region To find the center of mss of this lmin, prtition the intervl, into n suintervls of equl width Let i e the center of the ith suintervl You cn pproimte the portion of the lmin ling in the ith suintervl rectngle whose height is h f i g i Becuse the densit of the rectngle is, its mss is Densit Height Width Now, considering this mss to e locted t the center i, i of the rectngle, the directed distnce from the -is to i, i is i f i g i So, the moment of out the -is is m i m densitre A m i densitre f i g i Moment mssdistnce m i i f g d f i g i f i g i Summing the moments nd tking the limit s n suggest the definitions elow Moments nd Center of Mss of Plnr Lmin Let f nd g e continuous functions such tht f g on,, nd consider the plnr lmin of uniform densit ounded the grphs of f, g, nd The moments out the - nd -es re M M f g f g d f g d The center of mss is given M nd M, where m m, m is the mss of the lmin f g d

56 SECTION 76 Moments, Centers of Mss, nd Centroids 5 EXAMPLE The Center of Mss of Plnr Lmin Find the center of mss of the lmin of uniform densit f nd the -is ounded the grph of f() f() Figure 76 Editle Grph f() = Center of mss: 8, ( ) = The center of mss is the lncing point Figure 76 5 Solution Becuse the center of mss lies on the is of smmetr, ou know tht Moreover, the mss of the lmin is m To find the moment out the -is, plce representtive rectngle in the region, s shown in Figure 76 The distnce from the -is to the center of this rectngle is Becuse the mss of the representtive rectngle is f ou hve i f M 56 5 nd is given M m d d 6 8 d So, the center of mss (the lncing point) of the lmin is, 8 5, s shown in Figure 76 Tr It Eplortion A Open Eplortion The densit in Emple is common fctor of oth the moments nd the mss, nd s such divides out of the quotients representing the coordintes of the center of mss So, the center of mss of lmin of uniform densit depends onl on the shpe of the lmin nd not on its densit For this reson, the point, Center of mss or centroid is sometimes clled the center of mss of region in the plne, or the centroid of the region In other words, to find the centroid of region in the plne, ou simpl ssume tht the region hs constnt densit of nd compute the corresponding center of mss

57 5 CHAPTER 7 Applictions of Integrtion EXAMPLE 5 The Centroid of Plne Region f() = g() = + Find the centroid of the region ounded the grphs of g f nd f() + g() (, ) Figure 76 Editle Grph f() g() Hold pencil verticll nd move the oject on the pencil point until the centroid is locted Divide the oject into representtive elements Mke the necessr mesurements nd numericll pproimte the centroid Compre our result with the result in prt () (, ) EXPLORATION Cut n irregulr shpe from piece of crdord Solution The two grphs intersect t the points, nd,, s shown in Figure 76 So, the re of the region is A 9 A 9 9 f g d d 9 The centroid, of the region hs the following coordintes A d 9 d d 6 d 9 d So, the centroid of the region is,, 5 Tr It Eplortion A 5 For simple plne regions, ou m e le to find the centroids without resorting to integrtion EXAMPLE 6 The Centroid of Simple Plne Region () Originl region (, ) (, ) 5 (5, ) 5 6 () The centroids of the three rectngles Figure 765 Find the centroid of the region shown in Figure 765() Solution B superimposing coordinte sstem on the region, s shown in Figure 765(), ou cn locte the centroids of the three rectngles t,, nd Using these three points, ou cn find the centroid of the region A re of region 5,, 5, 5 5 So, the centroid of the region is (9, ) Tr It Eplortion A 9 9 NOTE In Emple 6, notice tht (9, ) is not the verge of 5,,,, nd 5,

58 SECTION 76 Moments, Centers of Mss, nd Centroids 5 R Centroid of R r ra The volume V is, where A is the re of region R Figure 766 Rottle Grph L Theorem of Pppus The finl topic in this section is useful theorem credited to Pppus of Alendri (c AD), Greek mthemticin whose eight-volume Mthemticl Collection is record of much of clssicl Greek mthemtics The proof of this theorem is given in Section THEOREM 7 The Theorem of Pppus Let R e region in plne nd let L e line in the sme plne such tht L does not intersect the interior of R, s shown in Figure 766 If r is the distnce etween the centroid of R nd the line, then the volume V of the solid of revolution formed revolving R out the line is V ra where A is the re of R (Note tht r is the distnce trveled the centroid s the region is revolved out the line) The Theorem of Pppus cn e used to find the volume of torus, s shown in the following emple Recll tht torus is doughnut-shped solid formed revolving circulr region out line tht lies in the sme plne s the circle (ut does not intersect the circle) EXAMPLE 7 Finding Volume the Theorem of Pppus Find the volume of the torus shown in Figure 767(), which ws formed revolving the circulr region ounded out the -is, s shown in Figure 767() ( ) + = r = (, ) Centroid () Figure 767 Torus () EXPLORATION Use the shell method to show tht the volume of the torus is given V d Evlute this integrl using grphing utilit Does our nswer gree with the one in Emple 7? Rottle Grph Solution In Figure 767(), ou cn see tht the centroid of the circulr region is, So, the distnce etween the centroid nd the is of revolution is r Becuse the re of the circulr region is A, the volume of the torus is V ra 95 Tr It Eplortion A Eplortion B

59 5 CHAPTER 7 Applictions of Integrtion Eercises for Section 76 The smol Click on Click on indictes n eercise in which ou re instructed to use grphing technolog or smolic computer lger sstem to view the complete solution of the eercise to print n enlrged cop of the grph In Eercises, find the center of mss of the point msses ling on the -is 5 Grphicl Resoning () Trnslte ech point mss in Eercise to the right five units nd determine the resulting center of mss () Trnslte ech point mss in Eercise to the left three units nd determine the resulting center of mss 6 Conjecture Use the result of Eercise 5 to mke conjecture out the chnge in the center of mss tht results when ech point mss is trnslted k units horizontll Sttics Prolems In Eercises 7 nd 8, consider em of length L with fulcrum feet from one end (see figure) There re ojects with weights W nd W plced on opposite ends of the em Find such tht the sstem is in equilirium 7 Two children weighing 5 pounds nd 75 pounds re going to pl on seesw tht is feet long 8 In order to move 55-pound rock, person weighing pounds wnts to lnce it on em tht is 5 feet long In Eercise 9, find the center of mss of the given sstem of point msses 9 m 6, m, m 5 5,, m 7, m, m, m 8,, 5, 6 m, m, m, m, m 5 7, 8,, 5, 5 8 m, m, m 6, m, m 5 6,,,, 5 8 W W m i, m i, 5,,, 5, 5, 5, L In Eercises, find M, M, nd, for the lmins of uniform densit ounded the grphs of the equtions In Eercises 5 8, set up nd evlute the integrls for finding the re nd moments out the - nd -es for the region ounded the grphs of the equtions (Assume ) In Eercises 9, use grphing utilit to grph the region ounded the grphs of the equtions Use the integrtion cpilities of the grphing utilit to pproimte the centroid of the region 9 m i, m i, m i,,,,,,,,,,, 8,,,,,,,,,,, 5, e,,, Prefricted End Section of Building 5, Witch of Agnesi 6 7,,, 6 5,, 5, 5, 5 5 6, 8 8,,,,

60 SECTION 76 Moments, Centers of Mss, nd Centroids 55 In Eercises 8, find nd/or verif the centroid of the common region used in engineering Tringle Show tht the centroid of the tringle with vertices,,,, nd, c is the point of intersection of the medins (see figure) (, c) (, c) ( +, c) 9 Grphicl Resoning Consider the region ounded the grphs of nd, where > () Sketch grph of the region () Use the grph in prt () to determine Eplin (c) Set up the integrl for finding M Becuse of the form of the integrnd, the vlue of the integrl cn e otined withou integrting Wht is the form of the integrnd nd wht is the vlue of the integrl? Compre with the result in prt () (d) Use the grph in prt () to determine whether > or < Eplin (, ) (, ) Figure for Figure for Prllelogrm Show tht the centroid of the prllelogrm with vertices,,,,, c, nd, c is the point of intersection of the digonls (see figure) 5 Trpezoid Find the centroid of the trpezoid with vertices,,,, c,, nd c, Show tht it is the intersection of the line connecting the midpoints of the prllel sides nd the line connecting the etended prllel sides, s shown in the figure (, ) (, ) Figure for 5 Figure for 6 6 Semicircle Find the centroid of the region ounded the grphs of r nd (see figure) 7 Semiellipse Find the centroid of the region ounded the grphs of nd (see figure) (c, ) (c, ) (, ) Figure for 7 Figure for 8 Prolic spndrel (, ) = 8 Prolic Spndrel Find the centroid of the prolic spndrel shown in the figure r r (, ) r (e) Use integrtion to verif our nswer in prt (d) Grphicl nd Numericl Resoning Consider the region ounded the grphs of n nd, where > nd n is positive integer () Set up the integrl for finding M Becuse of the form of the integrnd, the vlue of the integrl cn e otined without integrting Wht is the form of the integrnd nd wht is the vlue of the integrl? Compre with the result in prt () () Is > or < Eplin? (c) Use integrtion to find s function of n (d) Use the result of prt (c) to complete the tle (e) Find lim n (f) Give geometric eplntion of the result in prt (e) Modeling Dt The mnufcturer of glss for window in conversion vn needs to pproimte its center of mss A coordinte sstem is superimposed on prototpe of the glss (see figure) The mesurements (in centimeters) for the right hlf of the smmetric piece of glss re shown in the tle n 9 6 () Use Simpson s Rule to pproimte the center of mss of the glss () Use the regression cpilities of grphing utilit to find fourth-degree polnomil model for the dt (c) Use the integrtion cpilities of grphing utilit nd the model to pproimte the center of mss of the glss Compre with the result in prt ()

61 56 CHAPTER 7 Applictions of Integrtion Modeling Dt The mnufcturer of ot needs to pproimte the center of mss of section of the hull A coordinte sstem is superimposed on prototpe (see figure) The mesurements (in feet) for the right hlf of the smmetric prototpe re listed in the tle () Use Simpson s Rule to pproimte the center of mss of the hull section () Use the regression cpilities of grphing utilit to find fourth-degree polnomil models for oth curves shown in the figure Plot the dt nd grph the models (c) Use the integrtion cpilities of grphing utilit nd the model to pproimte the center of mss of the hull section Compre with the result in prt () In Eercises 6, introduce n pproprite coordinte sstem nd find the coordintes of the center of mss of the plnr lmin (The nswer depends on the position of the coordinte sstem) l d d l 7 Find the center of mss of the lmin in Eercise if the circulr portion of the lmin hs twice the densit of the squre portion of the lmin 8 Find the center of mss of the lmin in Eercise if the squre portion of the lmin hs twice the densit of the circulr portion of the lmin In Eercises 9 5, use the Theorem of Pppus to find the volume of the solid of revolution 9 The torus formed revolving the circle 5 6 out the -is 5 The torus formed revolving the circle out the -is 5 The solid formed revolving the region ounded the grphs of,, nd out the -is 5 The solid formed revolving the region ounded the grphs of,, nd 6 out the -is Writing Aout Concepts 5 Let the point msses m, m,, m n e locted t,,,,, n, n Define the center of mss, 5 Wht is plnr lmin? Descrie wht is ment the center of mss, of plnr lmin 55 The centroid of the plne region ounded the grphs of f,,, nd is 5 6, 8 5 Is it possile to find the centroid of ech of the regions ounded the grphs of the following sets of equtions? If so, identif the centroid nd eplin our nswer () () (c) (d) f,,, f,,, f,,, f,,, 56 Stte the Theorem of Pppus nd nd nd nd In Eercises 57 nd 58, use the Second Theorem of Pppus which is stted s follows If segment of plne curve C is revolved out n is tht does not intersect the curve (ecept possil t its endpoints), the re S of the resulting surfce of revolution is given the product of the length of C times the distnce d trveled the centroid of C 57 A sphere is formed revolving the grph of r out the -is Use the formul for surfce re, S r, to find the centroid of the semicircle r 58 A torus is formed revolving the grph of out the -is Find the surfce re of the torus 59 Let n e constnt, nd consider the region ounded f n, the -is, nd Find the centroid of this region As n, wht does the region look like, nd where is its centroid? Putnm Em Chllenge 6 Let V e the region in the crtesin plne consisting of ll points, stisfing the simultneous conditions nd Find the centroid, of V This prolem ws composed the Committee on the Putnm Prize Competition The Mthemticl Assocition of Americ All rights reserved

62 SECTION 77 Fluid Pressure nd Fluid Force 57 Section 77 Fluid Pressure nd Fluid Force Find fluid pressure nd fluid force Fluid Pressure nd Fluid Force Swimmers know tht the deeper n oject is sumerged in fluid, the greter the pressure on the oject Pressure is defined s the force per unit of re over the surfce of od For emple, ecuse column of wter tht is feet in height nd inch squre weighs pounds, the fluid pressure t depth of feet of wter is pounds per squre inch* At feet, this would increse to 86 pounds per squre inch, nd in generl the pressure is proportionl to the depth of the oject in the fluid Definition of Fluid Pressure The pressure on n oject t depth h in liquid is Pressure P wh where w is the weight-densit of the liquid per unit of volume BLAISE PASCAL (6 66) Pscl is well known for his work in mn res of mthemtics nd phsics, nd lso for his influence on Leiniz Although much of Pscl s work in clculus ws intuitive nd lcked the rigor of modern mthemtics, he nevertheless nticipted mn importnt results MthBio Below re some common weight-densities of fluids in pounds per cuic foot Ethl lcohol 9 Gsoline Glcerin 786 Kerosene 5 Mercur 89 Sewter 6 Wter 6 When clculting fluid pressure, ou cn use n importnt (nd rther surprising) phsicl lw clled Pscl s Principle, nmed fter the French mthemticin Blise Pscl Pscl s Principle sttes tht the pressure eerted fluid t depth h is trnsmitted equll in ll directions For emple, in Figure 768, the pressure t the indicted depth is the sme for ll three ojects Becuse fluid pressure is given in terms of force per unit re P FA, the fluid force on sumerged horizontl surfce of re A is Fluid force F PA (pressure)(re) h The pressure t h is the sme for ll three ojects Figure 768 Rottle Grph * The totl pressure on n oject in feet of wter would lso include the pressure due to Erth s tmosphere At se level, tmospheric pressure is pproimtel 7 pounds per squre inch

63 58 CHAPTER 7 Applictions of Integrtion EXAMPLE Fluid Force on Sumerged Sheet Find the fluid force on rectngulr metl sheet mesuring feet feet tht is sumerged in 6 feet of wter, s shown in Figure The fluid force on horizontl metl sheet is equl to the fluid pressure times the re Figure 769 Rottle Grph d h( i ) c L( i ) Clculus methods must e used to find the fluid force on verticl metl plte Figure 77 Rottle Grph Solution Becuse the weight-densit of wter is 6 pounds per cuic foot nd the sheet is sumerged in 6 feet of wter, the fluid pressure is P 66 7 pounds per squre foot Becuse the totl re of the sheet is A squre feet, the fluid force is F PA 7 98 pounds This result is independent of the size of the od of wter The fluid force would e the sme in swimming pool or lke Tr It In Emple, the fct tht the sheet is rectngulr nd horizontl mens tht ou do not need the methods of clculus to solve the prolem Consider surfce tht is sumerged verticll in fluid This prolem is more difficult ecuse the pressure is not constnt over the surfce Suppose verticl plte is sumerged in fluid of weight-densit w (per unit of volume), s shown in Figure 77 To determine the totl force ginst one side of the region from depth c to depth d, ou cn sudivide the intervl c, d into n suintervls, ech of width Net, consider the representtive rectngle of width nd length L i, where i is in the ith suintervl The force ginst this representtive rectngle is F i wdepthre wh i L i P wh The force ginst n such rectngles is n F i w n h i L i i i pounds squre feet squre foot Eplortion A Note tht w is considered to e constnt nd is fctored out of the summtion Therefore, tking the limit s n suggests the following definition Definition of Force Eerted Fluid The force F eerted fluid of constnt weight-densit w (per unit of volume) ginst sumerged verticl plne region from c to d is F w lim n h i L i i d wc hl d where h is the depth of the fluid t nd L is the horizontl length of the region t

64 SECTION 77 Fluid Pressure nd Fluid Force 59 EXAMPLE Fluid Force on Verticl Surfce 5 ft 8 ft 6 ft () Wter gte in dm ft 6 6 h() = (, ) (, 9) () The fluid force ginst the gte Figure 77 A verticl gte in dm hs the shpe of n isosceles trpezoid 8 feet cross the top nd 6 feet cross the ottom, with height of 5 feet, s shown in Figure 77() Wht is the fluid force on the gte when the top of the gte is feet elow the surfce of the wter? Solution In setting up mthemticl model for this prolem, ou re t liert to locte the - nd -es in severl different ws A convenient pproch is to let the -is isect the gte nd plce the -is t the surfce of the wter, s shown in Figure 77() So, the depth of the wter t in feet is Depth To find the length L of the region t, find the eqution of the line forming the right side of the gte Becuse this line psses through the points, 9 nd,, its eqution is In Figure 77() ou cn see tht the length of the region t is Length h L Finll, integrting from 9to, ou cn clculte the fluid force to e d F w hl d c 6 d d ,96 pounds Tr It Eplortion A Open Eplortion NOTE In Emple, the -is coincided with the surfce of the wter This ws convenient, ut ritrr In choosing coordinte sstem to represent phsicl sitution, ou should consider vrious possiilities Often ou cn simplif the clcultions in prolem locting the coordinte sstem to tke dvntge of specil chrcteristics of the prolem, such s smmetr

65 5 CHAPTER 7 Applictions of Integrtion EXAMPLE Fluid Force on Verticl Surfce 8 Oservtion window The fluid force on the window Figure A circulr oservtion window on mrine science ship hs rdius of foot, nd the center of the window is 8 feet elow wter level, s shown in Figure 77 Wht is the fluid force on the window? Solution To tke dvntge of smmetr, locte coordinte sstem such tht the origin coincides with the center of the window, s shown in Figure 77 The depth t is then Depth The horizontl length of the window is, nd ou cn use the eqution for the circle,, to solve for s follows Length Finll, ecuse rnges from to, nd using 6 pounds per cuic foot s the weight-densit of sewter, ou hve d F w hl d c 6 8 d Initill it looks s if this integrl would e difficult to solve However, if ou rek the integrl into two prts nd ppl smmetr, the solution is simple F 66 The second integrl is (ecuse the integrnd is odd nd the limits of integrtion re smmetric to the origin) Moreover, recognizing tht the first integrl represents the re of semicircle of rdius, ou otin F 66 5 h 8 L 685 pounds d 6 6 So, the fluid force on the window is 685 pounds d Tr It Eplortion A 5 f is not differentile t ± Figure 77 5 TECHNOLOGY To confirm the result otined in Emple, ou might hve considered using Simpson s Rule to pproimte the vlue of 8 8 d From the grph of f 8 however, ou cn see tht f is not differentile when ± (see Figure 77) This mens tht ou cnnot ppl Theorem 9 from Section 6 to determine the potentil error in Simpson s Rule Without knowing the potentil error, the pproimtion is of little vlue Use grphing utilit to pproimte the integrl

66 SECTION 77 Fluid Pressure nd Fluid Force 5 Eercises for Section 77 The smol Click on Click on indictes n eercise in which ou re instructed to use grphing technolog or smolic computer lger sstem to view the complete solution of the eercise to print n enlrged cop of the grph Force on Sumerged Sheet In Eercises nd, the re of the top side of piece of sheet metl is given The sheet metl is sumerged horizontll in 5 feet of wter Find the fluid force on the top side squre feet 6 squre feet Fluid Force of Wter In Eercises, find the fluid force on the verticl plte sumerged in wter, where the dimensions re given in meters nd the weight-densit of wter is 98 newtons per cuic meter Squre Squre Buont Force In Eercises nd, find the uont force of rectngulr solid of the given dimensions sumerged in wter so tht the top side is prllel to the surfce of the wter The uont force is the difference etween the fluid forces on the top nd ottom sides of the solid h h ft ft ft ft 6 ft 8 ft Tringle Rectngle Rottle Grph Fluid Force on Tnk Wll In Eercises 5, find the fluid force on the verticl side of the tnk, where the dimensions re given in feet Assume tht the tnk is full of wter 5 Rectngle 6 Tringle Rottle Grph 9 6 Force on Concrete Form In Eercises 5 8, the figure is the verticl side of form for poured concrete tht weighs 7 pounds per cuic foot Determine the force on this prt of the concrete form 5 7 Trpezoid 8 Semicircle 5 Rectngle 6 Semiellipse, ft ft 6 ft ft 9 Prol, Semiellipse, 7 Rectngle 8 Tringle 6 9 ft 5 ft ft 6 ft 9 Fluid Force of Gsoline A clindricl gsoline tnk is plced so tht the is of the clinder is horizontl Find the fluid force on circulr end of the tnk if the tnk is hlf full, ssuming tht the dimeter is feet nd the gsoline weighs pounds per cuic foot

67 5 CHAPTER 7 Applictions of Integrtion Fluid Force of Gsoline Repet Eercise 9 for tnk tht is full (Evlute one integrl geometric formul nd the other oserving tht the integrnd is n odd function) Fluid Force on Circulr Plte A circulr plte of rdius r feet is sumerged verticll in tnk of fluid tht weighs w pounds per cuic foot The center of the circle is k k > r feet elow the surfce of the fluid Show tht the fluid force on the surfce of the plte is F wk r (Evlute one integrl geometric formul nd the other oserving tht the integrnd is n odd function) Fluid Force on Circulr Plte Use the result of Eercise to find the fluid force on the circulr plte shown in ech figure Assume the pltes re in the wll of tnk filled with wter nd the mesurements re given in feet () Fluid Force on Rectngulr Plte A rectngulr plte of height h feet nd se feet is sumerged verticll in tnk of fluid tht weighs w pounds per cuic foot The center is k feet elow the surfce of the fluid, where h k Show tht the fluid force on the surfce of the plte is F wkh Fluid Force on Rectngulr Plte Use the result of Eercise to find the fluid force on the rectngulr plte shown in ech figure Assume the pltes re in the wll of tnk filled with wter nd the mesurements re given in feet () Sumrine Porthole A porthole on verticl side of sumrine (sumerged in sewter) is squre foot Find the fluid force on the porthole, ssuming tht the center of the squre is 5 feet elow the surfce 6 Sumrine Porthole Repet Eercise 5 for circulr porthole tht hs dimeter of foot The center is 5 feet elow the surfce () () Modeling Dt The verticl stern of ot with superimposed coordinte sstem is shown in the figure The tle shows the width w of the stern t indicted vlues of Find the fluid force ginst the stern if the mesurements re given in feet w Wter level 6 8 Irrigtion Cnl Gte The verticl cross section of n irrigtion cnl is modeled f 5 where is mesured in feet nd corresponds to the center of the cnl Use the integrtion cpilities of grphing utilit to pproimte the fluid force ginst verticl gte used to stop the flow of wter if the wter is feet deep In Eercises 9 nd, use the integrtion cpilities of grphing utilit to pproimte the fluid force on the vertic plte ounded the -is nd the top hlf of the grph of the eqution Assume tht the se of the plte is feet eneth the surfce of the wter 9 Think Aout It Stern 8 6 () Approimte the depth of the wter in the tnk in Eercise 5 if the fluid force is one-hlf s gret s when the tnk is full () Eplin wh the nswer in prt () is not 5 Writing Aout Concepts Define fluid pressure Define fluid force ginst sumerged verticl plne region Two identicl semicirculr windows re plced t the sme depth in the verticl wll of n qurium (see figure) Which hs the greter fluid force? Eplin d 6 w d 7

68 REVIEW EXERCISES 5 Review Eercises for Chpter 7 The smol Click on Click on indictes n eercise in which ou re instructed to use grphing technolog or smolic computer lger sstem to view the complete solution of the eercise to print n enlrged cop of the grph In Eercises, sketch the region ounded the grphs of the equtions, nd determine the re of the region csc, (one region) 9 In Eercises, use grphing utilit to grph the region ounded the grphs of the functions, nd use the integrtion cpilities of the grphing utilit to find the re of the region, In Eercises 5 8, use verticl nd horizontl representtive rectngles to set up integrls for finding the re of the region ounded the grphs of the equtions Find the re of the region evluting the esier of the two integrls ,,, 5,, 5,,,,,,, e, e, sin, cos, cos,,, 9 Think Aout It A person hs two jo offers The strting slr for ech is $,, nd fter ers of service ech will p $56, The slr increses for ech offer re shown in the figure From strictl monetr viewpoint, which is the etter offer? Eplin Slr (in dollrs) 6,,, S 8, 8,,,,,,,,,, Jo 5 7 Jo Yer t Modeling Dt The tle shows the nnul service revenue in illions of dollrs for the cellulr telephone industr for the ers 995 through (Source: Cellulr Telecommunictions & Internet Assocition) () Use the regression cpilities of grphing utilit to find n eponentil model for the dt Let t represent the er, with t 5 corresponding to 995 Use the grphing utilit to plo the dt nd grph the model in the sme viewing window () A finncil consultnt elieves tht model for service revenue for the ers 5 through is Wht is the difference in totl service revenue etween the two models for the ers 5 through? In Eercises 8, find the volume of the solid generted revolving the plne region ounded the equtions out the indicted line(s) () the -is () the -is (c) the line (d) the line 6 () the -is () the line (c) the -is (d) the line () the -is (olte spheroid) 6 9 () the -is (prolte spheroid) () the -is (olte spheroid) () the -is (prolte spheroid) ,,,, Yer R,,, revolved out the -is R 5 68e t,,, revolved out the -is,,, 6 revolved out the -is e,,, revolved out the -is In Eercises 9 nd, consider the region ounded the grphs of the equtions nd 9 Are Find the re of the region Volume Find the volume of the solid generted revolving the region out () the -is nd () the -is R

69 5 CHAPTER 7 Applictions of Integrtion Depth of Gsoline in Tnk A gsoline tnk is n olte spheroid generted revolving the region ounded the grph of 6 9 out the -is, where nd re mesured in feet Find the depth of the gsoline in the tnk when it is filled to one-fourth its cpcit Mgnitude of Bse The se of solid is circle of rdius, nd its verticl cross sections re equilterl tringles The volume of the solid is cuic meters Find the rdius of the circle In Eercises nd, find the rc length of the grph of the function over the given intervl f 6 5 5,,, 5 Length of Ctenr A cle of suspension ridge forms ctenr modeled the eqution cosh 8, where nd re mesured in feet Use grphing utilit to pproimte the length of the cle 6 Approimtion Determine which vlue est pproimtes the length of the rc represented the integrl sec d (Mke our selection on the sis of sketch of the rc nd not performing n clcultions) () () (c) (d) (e), 7 Surfce Are Use integrtion to find the lterl surfce re of right circulr cone of height nd rdius 8 Surfce Are The region ounded the grphs of,, nd is revolved out the -is Find the surfce re of the solid generted 9 Work A force of pounds is needed to stretch spring inch from its nturl position Find the work done in stretching the spring from its nturl length of inches to length of 5 inches Work The force required to stretch spring is 5 pounds Find the work done in stretching the spring from its nturl length of 9 inches to doule tht length Work A wter well hs n eight-inch csing (dimeter) nd is 75 feet deep The wter is 5 feet from the top of the well Determine the mount of work done in pumping the well dr, ssuming tht no wter enters it while it is eing pumped Work Repet Eercise, ssuming tht wter enters the well t rte of gllons per minute nd the pump works t rte of gllons per minute How mn gllons re pumped in this cse? Work A chin feet long weighs 5 pounds per foot nd is hung from pltform feet ove the ground How much work is required to rise the entire chin to the -foot level? Work A windlss, feet ove ground level on the top of uilding, uses cle weighing pounds per foot Find the work done in winding up the cle if () one end is t ground level () there is -pound lod ttched to the end of the cle 5 Work The work done vrile force in press is 8 footpounds The press moves distnce of feet nd the force is qudrtic of the form F Find 6 Work Find the work done the force F shown in the figure In Eercises 7 5, find the centroid of the region ounded the grphs of the equtions Pounds,,,,, 5 Centroid A lde on n industril fn hs the configurtion of semicircle ttched to trpezoid (see figure) Find the centroid of the lde 8 6 F 6 8 Feet 5 7 (9, ) 5 Fluid Force A swimming pool is 5 feet deep t one end nd feet deep t the other, nd the ottom is n inclined plne The length nd width of the pool re feet nd feet If the pool is full of wter, wht is the fluid force on ech of the verticl wlls? 5 Fluid Force Show tht the fluid force ginst n vertic region in liquid is the product of the weight per cuic volume of the liquid, the re of the region, nd the depth of the centroid of the region 5 Fluid Force Using the result of Eercise 5, find the fluid force on one side of verticl circulr plte of rdius feet th is sumerged in wter so tht its center is 5 feet elow the surfce

70 PS Prolem Solving 55 PS Prolem Solving The smol Click on Click on indictes n eercise in which ou re instructed to use grphing technolog or smolic computer lger sstem to view the complete solution of the eercise to print n enlrged cop of the grph Let R e the re of the region in the first qudrnt ounded the prol nd the line c, c > Let T e the re of the tringle AOB Clculte the limit lim T c R 5 A hole is cut through the center of sphere of rdius r (see figure) The height of the remining sphericl ring is h Find the volume of the ring nd show tht it is independent of the rdius of the sphere c A B(c, c ) h r T = R O c Rottle Grph Let R e the region ounded the prol nd the -is Find the eqution of the line m tht divides this region into two regions of equl re () A torus is formed revolving the region ounded the circle out the -is (see figure) Use the disk method to clculte the volume of the torus () Use the disk method to find the volume of the generl torus if the circle hs rdius r nd its center is R units from the is of rottion Grph the curve 8 = r = Use computer lger sstem to find the surfce re of the solid of revolution otined revolving the curve out the -is ( ) + = (, ) Centroid = m 6 A rectngle R of length l nd width w is revolved out the line L (see figure) Find the volume of the resulting solid of revolution L d Figure for 6 Figure for 7 7 () The tngent line to the curve t the point A, intersects the curve t nother point B Let R e the re of the region ounded the curve nd the tngent line The tngent line t B intersects the curve t nother point C (see figure) Let S e the re of the region ounded the curve nd this second tngent line How re the res R nd S relted? () Repet the construction in prt () selecting n ritrr point A on the curve Show tht the two res R nd S re lws relted in the sme w 8 The grph of f psses through the origin The rc length of the curve from, to, f is given s e t dt Identif the function f 9 Let f e rectifile on the intervl,, nd let s ft dt () Find ds d () Find ds nd ds (c) If f t t, find s on, R (d) Clculte s nd descrie wht it signifies w B R S C = A(, )

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