Paul s Notes. Chapter Planning Guide

Transcription

1 Applictions of Integrtion. Are of Region Between Two Curves. Volume: The Disk nd Wsher Methods. Volume: The Shell Method. Arc Length nd Surfces of Revolution Roof Are (Eercise, p. ) Sturn (Section Project, p. 7) Humer Bridge (Eercise, p. 7) Wter Tower (Eercise, p. 7) Pul s Notes Chpter Overview Chpter presents students with dditionl opportunities to ppl the concept of integrtion. B the end of this chpter, AB course students should e le to find the re of region etween two curves nd use vrious methods to find the volume of solid of revolution. In ddition, Section. discusses methods of finding the rc lengths of curves nd the res of surfces of revolution. Finding the rc lengths of curves is tested on the AP Clculus BC Em, nd finding the res of surfces of revolution, while on neither the AP Clculus AB nor BC Em, is eneficil ppliction for BC course students to lern. Students will hve plent of eposure to oth setting up nd computing the integrls involved in these pplictions. It is importnt to note tht use of clcultor snt is not ccepted on the AP Em. So, students must ecome proficient in using proper nottion to write these integrls hnd. Throughout this chpter, stress tht students should tret ech prolem individull nd tke the time to recognize ech sitution. For instnce, it is importnt, conceptull, to understnd the ide nd theor ehind ech prticulr method for finding the volume of solid of revolution so ou cn determine which one pplies in given sitution. Clockwise from top left, istockphoto.com/wwing; istockphoto.com/northern-light; Pul Brennn/Shutterstock.com; jl7/shutterstock.com; NASA Building Design (Eercise 7, p. ) An instructionl video from Pul, including ke points nd concepts covered in the chpter, is ville t LrsonClculusforAP.com. 7 Chpter Plnning Guide Chpter Resources Complete s Mnul Techer s Resources Guide Interctive Technolog Aville on Instructor Compnion site Lecture videos Cognero Pulisher Testing Softwre 7

2 8 Chpter Applictions of Integrtion Pul s Notes Section Overview An instructionl video from Pul, including teching strtegies for the section, is ville t LrsonClculusforAP.com. Essentil Question How do ou find the re of region etween two curves? Tell students tht the will lern how to nswer this question using the formul for the re of rectngle nd integrtion. Lesson Motivtor Finding the re of region etween two curves, rther thn under single curve, llows students to consider mn different regions tht the previousl could not. Connecting Nottions In regulr prtition, s the numer of representtive rectngles increses, or s n pproches infinit, the width of ech rectngle decreses, so Δ pproches zero.. 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 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 [, ]. Also, the grphs of oth f nd g lie ove the -is, nd the grph of g lies elow the grph of f, s shown in Figure.. You 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.. Are of region etween f nd g Figure. g f = Are of region under f g f Are of region under g [ f () g()] d = f () d g() d To verif the resonleness of the result shown in Figure., ou cn prtition the intervl [, ] into n suintervls, ech of width Δ. Then, s shown in Figure., sketch representtive rectngle of width Δ nd height f ( ) g( ), where i is in the ith suintervl. The re of this representtive rectngle is ΔA i = (height)(width) = [ f ( i ) g( i )] Δ. B dding the res of the n rectngles nd tking the limit s Δ (n ), ou otin lim n n [f ( i ) g( i )] Δ. i= Becuse f nd g re continuous on [, ], f g is lso continuous on [, ] nd the limit eists. So, the re of the region is Are = lim n n = i= [f ( i ) g( i )] Δ [ f () g()] d. f( i ) Figure. Representtive rectngle Height: f( i ) g( i ) Width: i g f g f g( i ) = Figure. Region etween two curves = Connecting Nottions Recll from Section. tht Δ is the norm of the prtition. In regulr prtition, the sttements Δ nd n re equivlent. g f 8

3 Section. Are of Region Between Two Curves 9 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., 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 is summrized grphicll in Figure.. Notice in Figure. tht the height of representtive rectngle is f () g() regrdless of the reltive position of the -is. f() g() Figure. (, f()) (, g()) f g f() g() (, f()) (, g()) Representtive rectngles re used throughout this chpter in vrious pplictions of integrtion. A verticl rectngle (of width Δ) implies integrtion with respect to, wheres horizontl rectngle (of width Δ) implies integrtion with respect to. Finding the Are of Region Between Two Curves Find the re of the region ounded the grphs of = +, =, =, nd =. Let g() = nd f () = +. Then g() f () for ll in [, ], s shown in Figure.. So, the re of the representtive rectngle is ΔA = [f () g()] Δ = [( + ) ( )] Δ nd the re of the region is A = = [f () g()] d [( + ) ( )] d = [ + + ] = + + = 7. f g Insight Finding the re of region etween two curves is tested on oth the AP Clculus AB nd BC Ems. (, f()) (, g()) f() = + g() = Region ounded the grph of f, the grph of g, =, nd = Figure. Pul s Notes Teching Strtegies In generl, encourge students to ecome efficient using their clcultors to correctl set up nd evlute integrls when finding the re of region etween two curves. This skill will e ssessed on the AP Em. You should, however, discourge students from using clcultor snt, such s (Y Y ) d, when writing out their solutions. This snt will not ern students points on free-response question on the AP Em. Etr Emple Find the re of the region ounded the grphs of = +, =, =, nd =. 9

4 Pul s Notes Etr Emple Find the re of the region ounded the grphs of f () = nd g() =. Etr Emple Find the re of the region... π f() = sin g() = Chpter 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. A Region Ling Between Two Intersecting Grphs Find the re of the region ounded the grphs of f () = nd g() =. In Figure., 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. = Set f () equl to g(). + = Write in generl form. ( + )( ) = Fctor. = or Solve for. So, = nd =. Becuse g() f () for ll in the intervl [, ], the representtive rectngle hs n re of ΔA = [f () g()] Δ = [( ) ] Δ nd the re of the region is A = [( ) ] d = [ + ] = 9. 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.7. Find the re of one of these regions. Let g() = cos nd f () = sin. Then g() f () for ll in the intervl corresponding to the shded region in Figure.7. To find the two points of intersection on this intervl, set f () nd g() equl to ech other nd solve for. sin = cos Set f() equl to g(). sin = cos Divide ech side cos. tn = Trigonometric identit = π or π, π Solve for. So, = π nd = π. Becuse sin cos for ll in the intervl [π, π], the re of the region is π A = [sin cos ] d π π = [ cos sin ]π =. (, f()) (, g()) Region ounded the grph of f nd the grph of g Figure. (, g()) (, f()) g() = f() = g() = cos f() = sin One of the regions ounded the grphs of the sine nd cosine functions Figure.7

5 Section. Are of Region Between Two Curves To find the re of the region etween two curves tht intersect t more thn two points, first determine ll points of intersection. Then check to see which curve is ove the other in ech intervl determined these points, s shown in Emple. Curves Tht Intersect t More thn Two Points See LrsonClculusforAP.com for n interctive version of this tpe of emple. Find the re of the region etween the grphs of f () = nd g() = +. Begin setting f () nd g() equl to ech other nd solving for. This ields the -vlues t ll points of intersection of the two grphs. = + Set f () equl to g(). = Write in generl form. ( )( + ) = Fctor. =,, Solve for. So, the two grphs intersect when =,, nd. In Figure.8, notice tht g() f () on the intervl [, ]. The two grphs switch t the origin, however, nd f () g() on the intervl [, ]. So, ou need two integrls one for the intervl [, ] nd one for the intervl [, ]. A = = [f () g()] d + ( ) d + = [ ] = ( ) + ( + ) = [g() f ()] d ( + ) d + [ + ] Resoning In Emple, notice tht ou otin n incorrect result when ou integrte from to. Such integrtion produces [f () g()] d = ( ) d =. When 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 or A = A = [(top curve) (ottom curve)] d in vrile [(right curve) (left curve)] d in vrile Verticl rectngles Horizontl rectngles where (, ) nd (, ) re either djcent points of intersection of the two curves involved or points on the specified oundr lines. Alger Review For help on the lger in Emple, see Emple in the Chpter Alger Review on pge A. g() f() f() g() (, ) (, ) (, 8) 8 g() = + f() = On [, ], g() f (), nd on [, ], f () g(). Figure.8 Pul s Notes Etr Emple Find the re of the region etween the grphs of f () = nd g() =..7 Resoning When finding the re of the region etween two curves tht intersect t more thn two points, students m fil to ccount for the fct tht the grphs switch. To help students void this error, encourge them to lws grph the functions efore setting up the integrls. Teching Strtegies If the given functions re written implicitl, it m e n indictor tht horizontl representtive rectngles should e considered. If the functions re more esil solved for or if solving for results in function tht is more esil integrted, students should use horizontl representtive rectngles.

6 Pul s Notes Etr Emple Find the re of the region ounded the grphs of = + nd =. 9 Common Errors When using horizontl representtive rectngles, students often fil to recognize tht the integrl needs to e entirel in terms of. Stress to students tht this mens the limits of integrtion should e in terms of s well. Chpter Applictions of Integrtion Horizontl Representtive Rectngles Find the re of the region ounded the grphs of = nd = +. Consider g() = nd f () = +. These two curves intersect when = nd =, s shown in Figure.9. Becuse f () g() on this intervl, ou hve ΔA = [g() f ()] Δ = [( ) ( + )] Δ. So, the re is A = [( ) ( + )] d = ( + ) d = [ + ] = ( + ) ( 8 ) = 9. Alger Review For help on the lger in Emple, see Emple in the Chpter Alger Review on pge A. f() = + (, ) = (, ) = (, ) g() = (, ) = Horizontl rectngles (integrtion Verticl rectngles (integrtion with with respect to ) respect to ) Figure.9 Figure. In Emple, notice tht integrting with respect to, ou need onl one integrl. To integrte with respect to, ou would need two integrls ecuse the upper oundr chnges t =, s shown in Figure.. A = = [( ) + ] d + [ + ( ) ] d + ] ( + ) d ( ) d = [ ( ) [ ( ) ] = ( ) ( + ) () + ( ) = 9

7 Section. Are of Region Between Two Curves 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, the re formul in this section ws developed s follows. A = (height)(width) ΔA = [ f () g()] Δ A = [ f () g()] d 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. The re of the region is 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.. Pul s Notes Etr Emple Find the re of the region ounded the grph of = + + nd the -is. Descrie the integrtion s n ccumultion process. 9 8; You cn think of the integrtion s n ccumultion of the res of the rectngles formed s the representtive rectngle slides from = to =. Lesson Closer Consider the rectngle ounded the lines =, =, =, nd =, s shown. A point from within the rectngle is picked t rndom. Wht is the proilit tht the point chosen will lie in portion of the rectngle not in the shded region? A = ( ) d = A = ( ) d = A = ( ) d =.... Aout.% g() = + 9 f() = cos ( ) A = ( ) d = 9 A = Figure. ( ) d =

8 . See mrgin. Chpter Applictions of Integrtion Assignment Guide D : Es.,,, 7,,,, 7, 8,,,, 9,,, 9,,, 7, 8,,,, 8, 7, 78, 8 8. ( ) d. ( + ) d. ( + ) d. ( ) d. ( ) d. [( ) ( )] d Eercises See ClcCht.com for tutoril help nd worked-out solutions to odd-numered eercises. Writing Definite Integrl In Eercises, set up the definite integrl tht gives the re of the region.. See mrgin.. =. = + + = = = +. = = + + = 8. = ( ). = ( ) = = 7. Finding Region 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. See mrgin. [( ) ] d [( + ) ] d 8. ) ] π d. 9. [(. π [( ) ] d. (sec cos ) d ( ) d Find the Error In Eercises nd, let R nd S e the two regions enclosed the grphs of f () = + + nd g() = + +, s shown. Descrie nd correct the error in the sttement. f R g 9. The re of R is given A = [g() f ()] d = ( + ) d. See mrgin.. The sum of the res of regions R nd S is given A = S [ f () g()] d = ( + ) d. See mrgin. Think Aout It In Eercises nd, 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.). f () = +, g() = ( ) d () () (c) (d) (e) 8. f () =, g() = () () (c) (d) (e) Compring Methods In Eercises 7 nd 8, find the re of the region integrting () with respect to nd () with respect to. (c) Compre our results. Which method is simpler? In generl, will this method lws e simpler thn the other one? Wh or wh not? 7 8. See mrgin. 7. = 8. = = = Should hve sutrcted g from f; A = 7. () ( + ) d () (c) Integrting with respect to ; Answers will vr. 8. () () (c) Integrting with respect to ; Answers will vr.. See Additionl Answers eginning on pge AA.

9 . See mrgin. Section. Are of Region Between Two Curves Finding the Are of Region In Eercises 9, sketch the region ounded the grphs of the equtions nd find the re of the region. 9. See mrgin. 9. =, = +, =, =. = +, =, =, =. f () = +, g() = +. = + +, = +. =, =, =. =, =, =, =. f () = +, g() = +. f () =, g() = 7. f () =, g() = + 8. f () = ( ), g() = 9. f () = +, g() =, =, =. f () =, g() =, =. f () =, =, =, =. g() =, =, = Finding the Are of Region In Eercises 8, () use grphing utilit to grph the region ounded the grphs of the equtions, () find the re of the region nlticll, nd (c) use the integrtion cpilities of the grphing utilit to verif our results. 8.. f () = ( + ), g() = See mrgin.. =, =. f () =, g() =. f () = 9, g() = 9 7. f () = +, g() = 8. f () = +, =, Finding the Are of Region In Eercises 9, sketch the region ounded the grphs of the functions nd find the re of the region. 9. See mrgin. 9. f() = cos, g() = cos, π. f () = sin, g() = cos, π π Finding the Are of Region In Eercises 8, () 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. 8. See mrgin.. f () = sin + sin, =, π. f () = sin + cos, =, π 7. f () = e, =, 8. g() = ln, =, = Finding the Are of Region In Eercises 9, () 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. 9. =, =, =. = e, =, =, =. =, = cos. =, = + Integrtion s n Accumultion Process In Eercises, 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.. See mrgin.. F() = ( t + ) dt () F() () F() (c) F(). F() = ( t + ) dt () F() () F() (c) F() α. F(α) = cos πθ dθ () F( ) () F() (c) F ( ). F() = e d () F( ) () F() (c) F() 9. See mrgin. Finding Are In Eercises 7, use integrtion to find the re of the figure hving the given vertices. 7. (, ), (, ), (, ) 8. (, ), (, ), (, ) 9 9. (, ), (, ), (, ), (, ). (, ), (, ), (, ), (, ). Numericl Integrtion Estimte the surfce re of the golf green using the Trpezoidl Rule (, ) (, ) (, ) (, ) 9.. (, ) (, ) (, ) (, ). f () = sin, g() = tn, π π. f () = sec π π tn, g() = ( ) +, =. f () = e, =,. f () =, g() = + ft ft ft ft ft ft ft ft ft ft Answers will vr. Smple nswers: () Aout 9 ft () Aout ft. (, ) (, ) (, ) (, ).. 7. (, ) (, ) (, ) (, ) (, ) (, ) (, ) 8 (, ( 9 8. See Additionl Answers eginning on pge AA.

10 . See mrgin.. Answers will vr. Emple: + on [, ] [( ) ( + )] d =. Answers will vr. Emple: on [, ], on [, ] ( ) d =. () The integrl [v (t) v (t)] dt = mens tht the first cr trveled more meters thn the second cr etween nd seconds. The integrl [v (t) v (t)] dt = mens tht the first cr trveled more meters thn the second cr etween nd seconds. The integrl [v (t) v (t)] dt = mens tht the second cr trveled more meters thn the first cr etween nd seconds. () No; You do not know when oth crs strted or the initil distnce etween the crs. (c) The cr with velocit v is hed meters. (d) Cr is hed 8 meters.. () The re etween the two curves represents the difference etween the ccumulted deficit under the two plns. () Proposl is etter ecuse the cumultive deficit (the re under the curve) is less. 7. = 9 ( ). 8. = Answers will vr. Smple nswer:.... (, ) f() = (, )...8. Chpter Applictions of Integrtion. Numericl Integrtion Estimte the surfce re of the oil spill using the Trpezoidl Rule. Aout 8. mi 7. Answers will vr. Smple nswer: f ( ) = (, ) (, ) mi. mi. mi mi. mi mi. mi mi WRITING ABOUT CONCEPTS See. Are Between Curves The grphs of = nd mrgin. = + intersect t three points. However, the re etween the curves cn e found single integrl. Eplin wh, nd write n integrl for this re.. Using Smmetr 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. See mrgin.. Interpreting Integrls Two crs with velocities v nd v re tested on stright trck (in meters per second). Consider the following. [v (t) v (t)] dt = [v (t) v (t)] dt = [v (t) v (t)] dt = See mrgin. () Write verl interprettion of ech integrl. () Is it possile to determine the distnce etween the two crs when t = seconds? Wh or wh not? (c) Assume oth crs strt t the sme time nd plce. Which cr is hed when t = seconds? How fr hed is the cr? (d) Suppose Cr hs velocit v nd is hed of Cr meters when t = seconds. How fr hed or ehind is Cr when t = seconds?. HOW DO YOU SEE IT? A stte legislture is deting two proposls for eliminting the nnul udget deficits fter ers. The rte of decrese of the deficits for ech proposl is shown in the figure. See mrgin. () Wht does the re etween the two curves represent? () From the viewpoint of minimizing the cumultive stte deficit, which is the etter proposl? Eplin. jl7/ Shutterstock.com Deficit (in illions of dollrs) D Proposl Proposl 8 Yer t Dividing Region In Eercises 7 nd 8, find such tht the line = divides the region ounded the grphs of the two equtions into two regions of equl re. 7. = 9, = 7 8. See mrgin. 8. = 9, = Dividing Region In Eercises 9 nd 7, find such tht the line = divides the region ounded the grphs of the equtions into two regions of equl re. 9. =, =, = =.7 7. =, = =.8 Limits nd Integrls In Eercises 7 nd 7, evlute the limit nd sketch the grph of the region whose re is represented the limit See mrgin. 7. lim Δ i=( n i i ) Δ, where i = i n nd Δ = n 7. lim Δ n ( i ) Δ, where i = + i i= n nd Δ = n 7. BUILDING DESIGN Concrete sections for new uilding hve the dimensions (in meters) nd shpe shown in the figure. (., ) = + = m (., ) () Find the re of the fce of the section superimposed on the rectngulr coordinte sstem. Aout. m () Find the volume of concrete in one of the sections multipling the re in prt () meters. Aout. m (c) One cuic meter of concrete weighs pounds. Find the weight of the section., l 7. Profit The chief finncil officer of compn reports tht profits for the pst fiscl er were $.9 million. The officer predicts tht profits for the net ers will grow t continuous nnul rte somewhere etween % nd %. Estimte the cumultive difference in totl profit over the ers sed on the predicted rnge of growth rtes. $. million

11 . See mrgin. Section. Are of Region Between Two Curves 7 7. 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. See mrgin. (c) Grph the model =. How does this model compre with the model in prt ()? See mrgin. (d) Use the integrtion cpilities of grphing utilit to pproimte the income inequlit. Aout.7 7. Mechnicl Design The surfce of mchine prt is the region etween the grphs of = nd =.8 + k (see figure). () Find k where the prol is tngent to the grph of. k =. () Find the re of the surfce of the mchine prt Are Find the re etween the grph of = sin nd the line segment joining the points (, ) nd (7π, ), s shown in the figure. See mrgin. (, ) 7, = c Figure for 77 Figure for 78 = 78. Are The horizontl line = c intersects the grph of = in the first qudrnt, s shown in the figure. Find c so tht the res of the two shded regions re equl. c = True or Flse? In Eercises 79 nd 8, determine whether the sttement is true or flse. If it is flse, eplin wh or give n emple tht shows it is flse. 79. If the grphs of f nd g intersect midw etween = nd =, then [ f () g()] d =. See mrgin. 8. The line = (.) divides the region under the curve f () = ( ) on [, ] into two regions of equl re. True Clculus AP Em Preprtion Questions 8. Multiple Choice The figure elow shows the grphs of = nd =. Wht is the re of the shded region? C (A) (B) (C) (D) 8 8. Multiple Choice Wht is the re of the region ounded the grphs of = nd = +? C (A) (B) (C) 9 (D) 8. Multiple Choice Which integrl gives the re A of the region ounded the grph of f () = nd the tngent line to the grph of f t (, )? D (A) A = ( + + ) d (B) A = ( ) d (C) A = ( + + ) d (D) A = ( + + ) d 7. () (c) Percents of totl income Percents of totl income Percents of fmilies 8 Percents of fmilies π Flse; Let f () = nd g() =. f nd g intersect t (, ), the midpoint of [, ], ut [ f () g()] d [ ( )] d = =. 7

12 8 Chpter Applictions of Integrtion Pul s Notes Section Overview An instructionl video from Pul, including teching strtegies for the section, is ville t LrsonClculusforAP.com.. Volume: The Disk nd Wsher Methods 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 You hve lred lerned 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 three-dimensionl 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.. Essentil Question How cn ou use integrls to find the volume of solid? Tell students tht the will lern how to nswer this question looking t two methods for finding the volume of solid of revolution s well s method for finding the volume of solid with known cross section. Lesson Motivtor In this lesson, students lern how to ppl integrls to three-dimensionl prolems. Students will eplore different methods for finding the volumes of ojects with uncommon shpes tht re not esil defined well-known geometric formuls, such s funnels nd ottles. Solids of revolution Figure. When 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.. The volume of such disk is Volume of disk = (re of disk)(width of disk) = πr 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. out the indicted is, s shown on the net pge. 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 w Rectngle R Ais of revolution w Disk R ΔV = πr Δ. Approimting the volume of the solid n such disks of width Δ nd rdius R( i ) produces Volume of solid n π[r( i )] Δ i= = π n i= [R( i )] Δ. Volume of disk: πr w Figure. 8

13 Section. Volume: The Disk nd Wsher Methods 9 R Representtive rectngle Plne region Ais of revolution Representtive disk Pul s Notes = = Solid of revolution Approimtion n disks Disk method Figure. This pproimtion ppers to ecome etter nd etter s Δ (n ). So, ou cn define the volume of the solid s Volume of solid = lim π Δ n i= [R( i )] Δ = π [R()] d. Schemticll, the disk method looks like this. Known Preclculus Representtive New Integrtion Formul Element Formul Volume of disk V = πr w ΔV = π[r( i )] Δ Solid of revolution V = π [R()] d A similr formul cn e derived when 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 formuls elow. (See Figure..) Horizontl Ais of Revolution Volume = V = π [R()] d R() V = [R()] d Horizontl is of revolution Figure. Verticl Ais of Revolution d Volume = V = π [R()] d c d c R() d V = c [R()] Verticl is of revolution d Resoning In Figure., note tht ou cn determine the vrile of integrtion plcing representtive rectngle in the plne region perpendiculr to the is of revolution. When the width of the rectngle is Δ, integrte with respect to, nd when the width of the rectngle is Δ, integrte with respect to. Teching Strtegies In generl, e sure tht students re proficient in setting up the integrls used in oth the disk method nd the wsher method. On the free-response section of the AP Em, points m e wrded simpl for correctl identifing the limits of integrtion nd identifing π s necessr constnt. Resoning Note tht the representtive rectngle is plced perpendiculr to the is of revolution so tht disk is formed when the rectngle is revolved round the is. It is importnt tht students mke the connection etween the plcement of the representtive rectngle nd the nme of the method. This will help them distinguish etween the methods s the re introduced. 9

14 Pul s Notes Common Errors There re mn common errors students cn mke when setting up n integrl to solve prolem using the disk method, such s the following. Forgetting to include π s constnt Forgetting to squre the rdius function Forgetting to include d or d in the setup Forgetting to sutrct the eqution for the is of revolution when defining the rdius function in cses where the is of revolution is not coordinte is Etr Emple Find the volume of the solid formed revolving the region ounded the grph of f () = + nd the -is out the -is. π Etr Emple Find the volume of the solid formed revolving the region ounded the grphs of =, =, nd = out the line =. 7π Chpter Applictions of Integrtion The simplest ppliction of the disk method involves plne region ounded the grph of f nd the -is. When the is of revolution is the -is, the rdius R() is simpl f (). Using the Disk Method Find the volume of the solid formed revolving the region ounded the grph of f () = sin nd the -is ( π) out the -is. From the representtive rectngle in the upper grph in Figure., ou cn see tht the rdius of this solid is R() = f () = sin. So, the volume of the solid of revolution is V = π π = π = π π [R()] d ( sin ) d sin d = π [ cos ] π = π(l + ) = π. Appl disk method. Sustitute sin for R(). Simplif. Integrte. Using Line Tht Is Not Coordinte Ais Find the volume of the solid formed revolving the region ounded the grphs of f () = nd g() = out the line =, s shown in Figure.7. 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() = ( ) = To find the volume, integrte etween nd. V = π = π = π [R()] d Appl disk method. ( ) d Sustitute for R(). ( + ) d Simplif. = π [ + = π ] Integrte. Figure. f() = Plne region Solid of revolution Figure.7 sin Plne region Solid of revolution f() = Ais of revolution f() g() = R() g() R()

15 Section. Volume: The Disk nd Wsher Methods The Wsher Method 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.8. If r nd R re the inner nd outer rdii of the wsher nd w is the width of the wsher, then the volume is Volume of wsher = π(r r )w. 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.9. If the region is revolved out its is of revolution, then the volume of the resulting solid is V = π ([R()] [r()] ) d. Wsher method Note tht the integrl involving the inner rdius represents the volume of the hole nd is sutrcted from the integrl involving the outer rdius. Figure.9 R() r() Plne region Solid of revolution with hole 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.. In Figure., ou cn see tht the outer nd inner rdii re s follows. R() = Outer rdius r() = Inner rdius Figure.8 R = Disk (, ) Ais of revolution R w w r Solid of revolution = Plne region r = = (, ) r R Pul s Notes Common Errors There re mn common errors students cn mke when setting up n integrl to solve prolem using the wsher method, such s the following. Forgetting to include π s constnt Forgetting to squre the rdius functions Forgetting to include d or d in the setup Forgetting to sutrct the eqution for the is of revolution when defining the rdius functions in cses where the is of revolution is not coordinte is Switching the order of R() nd r(), nd writing the integrnd s [r()] [R()] Etr Emple Find the volume of the solid formed revolving the region ounded the grphs of = nd = out the -is. π Integrting etween nd produces V = π = π = π = π [ = π. ([R()] [r()] ) d [( ) ( ) ]d ( ) d Simplif. ] Appl wsher method. Sustitute for R() nd for r(). Integrte. Solid of revolution Solid of revolution Figure.

16 Pul s Notes Common Errors Students often forget to integrte in terms of when the prolem involves verticl is of revolution. Etr Emple Find the volume of the solid formed revolving the region ounded the grphs of = +, =, =, nd = out the -is. 8π Teching Strtegies On the AP Em, students will not e told which method to ppl or which vrile to use s the vrile of integrtion. Provide students with the following steps to help them orgnize their solutions. Drw the region descried in the prolem. Determine which vrile to use s the vrile of integrtion. Determine which method to use identifing whether the solid hs hole. Identif the rdius, or rdii, of the solid. Identif the limits of integrtion. Set up the integrl, including π s constnt, nd solve. Chpter 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. 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.. For : R = r = For : R = r = Plne region Figure. r R (, ) Solid of revolution For the region shown in Figure., 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 = π ( ) d + π = π d + π = π [ ] + π [ [ ( ) ] d Appl wsher method. ( ) d Simplif. ] = π + π ( + ) Integrte. = π Note tht the first integrl π d represents the volume of right circulr clinder of rdius nd height. This portion of the volume could hve een determined without using clculus. Alger Review For help on the lger in Emple, see Emple () in the Chpter Alger Review on pge A. Technolog Some grphing utilities hve the cpilit of generting (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.. Figure. Generted Mthemtic

17 Section. Volume: The Disk nd Wsher Methods Mnufcturing See LrsonClculusforAP.com for n interctive version of this tpe of emple. A mnufcturer drills hole through the center of metl sphere of rdius inches, s shown in Figure.(). The hole hs rdius of inches. Wht is the volume of the resulting metl ring? You cn imgine the ring to e generted segment of the circle whose eqution is + =, s shown in Figure.(). Becuse the rdius of the hole is inches, ou cn let = nd solve the eqution + = to determine tht the limits of integrtion re =±. So, the inner nd outer rdii re r() = nd R() =, nd the volume is V = π = π = π ([R()] [r()] ) d [( ) () ] d ( ) d = π [ ] = π cuic inches. Solids with Known Cross Sections With the disk method, ou cn find the volume of solid hving circulr cross section whose re is A = πr. 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. Volumes of Solids with Known Cross Sections. For cross sections of re A() tken perpendiculr to the -is, Volume = A() d. See Figure.().. For cross sections of re A() tken perpendiculr to the -is, d Volume = A() d. See Figure.(). c = = () Cross sections perpendiculr to -is Figure. = c = d () Cross sections perpendiculr to -is () () Figure. Solid of revolution in. R() = = r() = Plne region = in. Alger Review For help on the lger in Emple, see Emple () in the Chpter Alger Review on pge A. Pul s Notes Etr Emple A mnufcturer drills hole through the center of metl hemisphere of rdius inches, s shown. The hole hs rdius of inch. Wht is the volume of the resulting metl ring? π in. in. in. Teching Strtegies Students often hve difficult visulizing the concept of using integrls to find the re of solid using its cross section. The following tips m help students etter understnd this method. To led into discussion out setting up the integrl used in this method, rrnge fom cross sections to uild models of solids with our students. This will llow the students to visulize the integrl. Encourge students to drw two-dimensionl digrm of the cross section to help them identif the correct re formul. The most common cross sections used on the AP Em re squres, rectngles, semicircles, nd equilterl tringles. Be sure students commit the re formuls for these shpes to memor.

18 Pul s Notes Etr Emple Find the volume of the solid shown. The se of the solid is the region ounded the lines = nd =. The cross sections perpendiculr to the -is re squres. Etr Emple 7 Prove tht the volume of right circulr cone is V = πr h, where h is the height of the cone nd r is the rdius of the se. V = = h h A() d π [ r h (h ) ] d = πr ) h [(h ] h = πr h (h ) = πr h Lesson Closer Determine whether the disk method or wsher method should e used to find the volume of ech solid formed revolving the given region out the given line. Then set up ech integrl. Chpter Applictions of Integrtion Tringulr Cross Sections Find the volume of the solid shown in Figure.. The se of the solid is the region ounded the lines f () =, g() = +, nd =. The cross sections perpendiculr to the -is re equilterl tringles. The se nd re of ech tringulr cross section re s follows. Bse = ( ) ( + ) = Length of se Are = (se) Are of equilterl tringle A() = ( ) Are of cross section Becuse rnges from to, the volume of the solid is A() d = ( ) d = [ V = An Appliction to Geometr ( ) ] =. 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. As shown in Figure., 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 = (h ) h where is the length of the sides of the se of the prmid. So, A() = ( ) = h (h ). Integrting etween nd h produces h V = A() d = h = h h h (h ) d (h ) d = ( ) h)[(h ] h = h ( h ) = hb. B = = g() Cross sections re equilterl tringles. Tringulr se in -plne Figure. Figure. f() = g() = + = f() Alger Review For help on the lger in Emple 7, see Emple in the Chpter Alger Review on pge A7. Are = A() Are of se = B = h = (h ) h h. = R R R. =. R, = Disk; V = π ( ) d. R, = Wsher; V = π [( ) ( ) ] d c. R, = Disk; V = π d. R, = Wsher; V = π ( ) d [ ( ) ] d

19 Section. Volume: The Disk nd Wsher Methods. Eercises See ClcCht.com for tutoril help nd worked-out solutions to odd-numered eercises. Finding the Volume of Solid In Eercises, set up nd evlute the integrl tht gives the volume of the solid formed 9. =. = + revolving the region out the -is... = +. = See mrgin.. =. = 9. =, =. =, = 7. See mrgin. Finding the Volume of Solid In Eercises 7, set up nd evlute the integrl tht gives the volume of the solid formed revolving the region out the -is. 7. = 8. = 7. 8π 8. π 7 9. π ( 8 ln 7 ) 8.8. π[8 ln( + ) ] 7. Find the Error Let A e the region enclosed the grphs of = e, =, nd =, s shown. In Eercises nd, descrie nd correct the error in setting up the given integrl, which gives the volume V of the solid formed revolving the region out the -is.. See mrgin.. V = π 8 A (e ) d. V = π [ ( ln ) ] d Finding the Volume of Solid In Eercises, find the volumes of the solids generted revolving the region ounded the grphs of the equtions out the given lines.. =, =, =. See mrgin. () -is () -is (c) = (d) =. =, =, = () -is () -is (c) = 8 (d) =. =, = () -is () =. = +, = () -is () = 7. See mrgin. Finding the Volume of Solid In Eercises 7, find the volume of the solid generted revolving the region ounded the grphs of the equtions out the line =. 7. =, =, = 8. =, =, = 9. = ( + ) =, =, =. = sec, =, π Assignment Guide Es.,,, 7,, 7, 9,, 8 even,,, even,,, 7 7. π ( + ) d = π. π ( ) d = π. π ( ) d = π. π ( 9 ) d = 8π. π [( ) ( ) ] d = π. π 7. π 8. π [( ) () ] d ( ) d = 8π = 8 π.9 ( ) d = 8π 9. π ( ) d = π. π ( + ) d = π. Should e integrting with respect to not ; V = e [ ( ln ) ] d. The limits of integrtion should e nd e. V =. () 9π e (c) π [ ( ln ) ] d () π. () π () 8π (c) 89π. () π. () π (d) 8π (d) π () π () 8π

20 . See mrgin.. π. 9π. 8π. π ( 7 ln ).7 9. π ( e).8. π(e ) π. π.9. π π (e ). 8. π(e + e + e e ) 9.. π. π. π. π 7. π 8. π 9. π. π. A sine curve on [, π ] revolved out the -is. The region ounded =, =, =, nd = revolved out the -is. The prol = is horizontl trnsltion of the prol =. Therefore, their volumes re equl. Chpter Applictions of Integrtion Finding the Volume of Solid In Eercises, find the volume of the solid generted revolving the region ounded the grphs of the equtions out the line =.. See mrgin.. =, =, =, =. =, =, =, =. =, =. =, =, =, = Finding the Volume of Solid In Eercises, find the volume of the solid generted revolving the region ounded the grphs of the equtions out the -is.. =, =, =, = π ln + 8π. =, = 7. = π, =, =, = 8. = π, =, =, = = e, =, =, = 9.. = e, =, =, = See mrgin.. = +, = + +, =, =. =, = +, =, = 8 8π Finding the Volume of Solid In Eercises nd, find the volume of the solid generted revolving the region ounded the grphs of the equtions out the -is.. = ( ), =, = 8π π. = 9, =, =, = Finding the Volume of Solid In Eercises 8, 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.. = sin, =, =, = π. = cos, =, =, = π 7. = e, =, =, = 8. = e + e, =, =, = 8. See mrgin. Finding the Volume of Solid In Eercises 9, 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. 9. = e, =, =, =.98. = ln, =, =, =.. = rctn(.), =, =, =.. =, =.99 Finding the Volume of Solid In Eercises, find the volume generted rotting the given region out the specified line.. See mrgin.. R = R R. =. R out =. R out =. R out =. R out = 7. R out = 8. R out = 9. R out =. R out = WRITING ABOUT CONCEPTS Descriing Solid In Eercises nd, the integrl represents the volume of solid. Descrie the solid. π.. π sin d. π d See mrgin.. Compring Volumes 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. See mrgin.. HOW DO YOU SEE IT? Use the grph to mtch the integrl for the volume with the is of rottion. () V = π () V = π (c) V = π (d) V = π = f() = f() ( [ f ()] ) d ii (i) -is ( [ f ()] ) d iv (ii) -is [ f ()] d i (iii) = [ f ()] d iii (iv) =

21 . See mrgin. Section. Volume: The Disk nd Wsher Methods 7 Dividing Solid In Eercises nd, consider. WATER TOWER the solid formed revolving the region ounded A tnk on wter =, =, nd = out the -is. tower is sphere. Find the vlue of in the intervl [, ] tht divides the of rdius feet. solid into two prts of equl volume. Determine the depths. Find the vlues of in the intervl [, ] tht divide the of the wter when solid into three prts of equl volume. See mrgin. the tnk is filled to one-fourth nd threefourths 7. Mnufcturing 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. See mrgin. of its cpcit. (Note: Use the zero or root feture of 8. Mnufcturing For the metl sphere in Eercise 7, grphing utilit fter let R =. Wht vlue of r will produce ring whose evluting the definite integrl.). ft volume is ectl hlf the volume of the sphere? See mrgin. 9. Volume of Cone Use the disk method to verif tht 7. Minimum Volume The rc of = ( ) on the volume of right circulr cone is π r h, where r is the intervl [, ] is revolved out the line = (see the rdius of the se nd h is the height. Proof figure). See mrgin.. Volume of Sphere Use the disk method to verif () Find the volume of the resulting solid s function tht the volume of sphere is π r, where r is the rdius. Proof of.. Using Cone A cone of height H with se of rdius r is cut plne prllel to nd h units ove () Use grphing utilit to grph the function in prt (), nd use the grph to pproimte the vlue of the se, where h < H. Find the volume of the solid tht minimizes the volume of the solid. (frustum of cone) elow the plne. See mrgin. (c) Use clculus to find the vlue of tht minimizes See. Using Sphere A sphere of rdius r is cut the volume of the solid, nd compre the result with mrgin. plne h units ove the equtor, where h < r. Find the the nswer to prt (). volume of the solid (sphericl segment) ove the plne.. 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, where nd re mesured in meters. Use grphing utilit to grph the function nd find the volume of the tnk. See mrgin.. Volume of L Glss A glss continer cn e modeled revolving the grph of = { ,.9, Pul Brennn/Shutterstock.com.. < out the -is, where nd re mesured in centimeters. Use grphing utilit to grph the function nd find the volume of the continer. See mrgin.. Finding Volumes of Solid Find the volumes of the solids (see figures) generted if the upper hlf of the ellipse 9 + = is revolved out () the -is to form prolte spheroid (shped like footll), nd () the -is to form n olte spheroid. () π () π Figure for () Figure for () 8. Think Aout It Mtch ech integrl with the solid whose volume it represents, nd give the dimensions of ech solid. See mrgin. () Right circulr clinder () Ellipsoid (c) Sphere (d) Right circulr cone (e) Torus h (i) π (r h) d h (ii) π r d r (iii) π ( r ) d r (iv) π ( ) r (v) π [(R + r ) (R r ) ] d r d. =, 7. V = π(r r ) 8. r = 8.. πr h ( h H + h ) H. π (r r h + h ).... π m 8 8 Aout.9 cm 7. () V = π( + () ).7 (c) = () ii; right circulr clinder of rdius r nd height h () iv; ellipsoid whose underling ellipse hs the eqution () + () = (c) iii; sphere of rdius r (d) i; right circulr cone of rdius r nd height h (e) v; torus of cross-sectionl rdius r nd other rdius R 7

22 . See mrgin. 7. () When = : + = represents squre. When = : + = represents circle. () Solve the eqution for, s shown elow. Then form n slices, ech of whose re is pproimted the integrl elow. Finll, sum the volumes of these n slices. = ( ) A = = ( ) d ( ) d 8 Chpter Applictions of Integrtion 9. Cvlieri s Theorem Prove tht if two solids hve equl ltitudes nd ll plne sections prllel to their ses nd t equl distnces from their ses hve equl res, then the solids hve the sme volume (see figure). Proof R R Are of R = re of R 7. Using Cross Sections Find the volumes of the solids whose ses re ounded the grphs of = + nd =, with the indicted cross sections tken perpendiculr to the -is. 8 9 () Squres () Rectngles of height 7. Using Cross Sections Find the volumes of the solids whose ses re ounded the circle + =, with the indicted cross sections tken perpendiculr to the -is. () Squres (c) Semicircles 8 π () Equilterl tringles (d) Isosceles right tringles h Clculus AP Em Preprtion Questions 7. Multiple Choice The region shown in the figure is revolved out the -is, the -is, nd the line =. Which of the following orders the volumes of the resulting solids from lest to gretest? C (A) -is, -is, = (B) -is, -is, = 8 = (C) -is, =, -is (D) =, -is, -is 7. Multiple Choice Let R e the region ounded the grphs of = nd = for. Wht is the volume of the solid generted when R is revolved out the line =? B (A) π [ ( ) ( ) ] d (B) π [( ) ( ) ] d (C) π (9 ) d 9 (D) π [( ) ( ) ] d 7. Free Response Let f nd g e the functions defined f () = + + nd g() = sin(π). Let A nd B e the two regions enclosed the grphs of f nd g shown in the figure. 7. Using Cross Sections 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. See mrgin. + = + = + = (, ) A B.9 () Find the sum of the res of regions A nd B. () Region B is the se of solid whose cross sections perpendiculr to the -is re squres. Find the volume of the solid.. (c) Let h e the verticl distnce etween the grphs of f nd g in region B. Find the rte t which h chnges with respect to when =...9 8

23 Section. Volume: The Shell Method 9. Volume: The Shell Method Find the volume of solid of revolution using the shell method. Compre the uses of the disk nd wsher methods nd the shell method. The Shell Method 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.7, 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. Outer rdius Inner rdius So, the volume of the shell is Volume of shell = (volume of clinder) (volume of hole) w p Figure.7 h p + w w p Ais of revolution Pul s Notes Section Overview An instructionl video from Pul, including teching strtegies for the section, is ville t LrsonClculusforAP.com. Essentil Question How is the shell method used to find the volume of solid of revolution? Tell students tht the will lern how to nswer this question through n introduction to the shell method nd compring this method to the methods discussed in Section.. = π ( p + w ) h π ( p w ) h = πphw = π(verge rdius)(height)(thickness). You cn use this formul to find the volume of solid of revolution. For instnce, the plne region in Figure.8 is revolved out line to form the indicted solid. Consider horizontl rectngle of width Δ. As the plne region is revolved out line prllel to the -is, the rectngle genertes representtive shell whose volume is ΔV = π[p()h()] Δ. You cn pproimte the volume of the solid n such shells of thickness Δ, height h( i ), nd verge rdius p( i ). Volume of solid n π[p( i )h( i )] Δ = π n [ p( i )h( i )] Δ d p() c h() Plne region Ais of revolution Lesson Motivtor B eploring n lterntive method used to find the volumes of solids of revolution, students will e le to choose from the vrious methods determining which one is most convenient sed on the given informtion. i= This pproimtion ppers to ecome etter nd etter s Δ (n ). So, the volume of the solid is Volume of solid = lim π Δ n [ p( i )h( i )] Δ i= d = π [ p()h()] d. c i= Figure.8 Solid of revolution 9

24 Chpter Applictions of Integrtion Pul s Notes Common Errors When setting up n integrl to use the shell method, students often mistkenl use π s the constnt. Remind them tht, lthough π is the constnt used in the disk nd wsher methods, the constnt used in the shell method is π. Similrl, students often forget to multipl p() in the integrnd. Etr Emple Find the volume of the solid of revolution formed revolving the region ounded = + nd the -is ( ) out the -is. π The Shell Method To find the volume of solid of revolution with the shell method, use one of the formuls elow. (See Figure.9.) d c h() Horizontl is of revolution Figure.9 p() p() Verticl is of revolution 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. Horizontl Ais of Revolution d Volume = V = π p()h() d c Becuse the is of revolution is verticl, use verticl representtive rectngle, s shown in Figure.. The width Δ 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, ppl the shell method to find the volume of the solid. V = π p()h() d = π ( ) d = π ( + ) d ] = π [ + = π ( + ) = π Simplif. Integrte. Verticl Ais of Revolution Volume = V = π p()h() d Ais of revolution Figure. = h() = p() = h() (, )

25 Section. Volume: The Shell Method Using the Shell Method to Find Volume Find the volume of the solid of revolution formed revolving the region ounded the grph of = e nd the -is ( ) out the -is. Pul s Notes Becuse the is of revolution is horizontl, use horizontl representtive rectngle, s shown in Figure.. The width Δ 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 = π p()h() d c = π e d = π [ e ] = π ( e).98. Appl shell method. Integrte. Eplortion 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, Wsher, nd Shell Methods The disk, wsher, nd shell methods cn e distinguished s follows. For the disk nd wsher methods, 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. The wsher method nd the shell method re compred in Figure.. p() = Figure. h() = e = e Ais of revolution Common Errors When trnsitioning etween the disk nd wsher methods nd the shell method, students m ecome confused out which vrile to use s the vrile of integrtion. Point out tht when ppling the shell method, the representtive rectngle is lws prllel to the is of revolution. This, in turn, determines the vrile of integrtion. For instnce, in Emple, the is of revolution is the -is, so the representtive rectngle is horizontl. This mens tht the integrl should e written in terms of. Etr Emple Find the volume of the solid of revolution formed revolving the region ounded = nd the -is ( ) out the -is. 8π d d V = c (R r ) d V = (R r ) d V = ph d d d V = ph d c r c R R r h p c p h Verticl is of revolution Horizontl is of revolution Wsher method: Representtive rectngle is perpendiculr to the is of revolution. Figure. Verticl is of revolution Horizontl is of revolution Shell method: Representtive rectngle is prllel to the is of revolution. Eplortion e V = π d + π ( ln ) d e e = π [ ] π [ ( + ln ) ] = π e π ( e ) = π ( e).98 e

26 Chpter Applictions of Integrtion Pul s Notes Often, one method is more convenient to use thn the other. The net emple illustrtes cse in which the shell method is preferle. Shell Method Preferle Etr Emple Use the shell method to find the volume of the solid formed revolving the region ounded the grphs of = +, =, =, nd = out the -is. 8π See LrsonClculusforAP.com for n interctive version of this tpe of emple. Find the volume of the solid formed revolving the region ounded the grphs of = +, =, =, nd = out the -is. In Emple in Section., ou sw tht the wsher method requires two integrls to determine the volume of this solid. See Figure.(). V = π = π = π [ ] ( ) d + π [ ( ) ] d Appl wsher method. d + π ( ) d Simplif. + π [ ] = π + π ( + ) Integrte. For : R = r = For : R = r = r Ais of revolution (, ) () Disk method = π In Figure.(), ou cn see tht the shell method requires onl one integrl to find the volume. V = π p()h() d = π ( + ) d = π ( + ) d Simplif. = π [ + ] = π ( ) Appl shell method. Integrte. h() = + () Shell method Figure. p() = Ais of revolution (, ) = π Consider the solid formed revolving the region in Emple 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 integrl V = π ( )( + ) d p() = which gives the volume of the solid.

27 Section. Volume: The Shell Method Volume of Pontoon A pontoon is to e mde in the shpe shown in Figure.. The pontoon is designed rotting the grph of =,, out the -is, where nd re mesured in feet. Find the volume of the pontoon. Refer to Figure. nd use the disk method s shown. V = π = π ( ) d Appl disk method. ( 8 + ) d Simplif. = π [ + 8] = π. cuic feet Integrte. Figure. 8 ft Disk method Figure. r() = R() = ft Pul s Notes Etr Emple A pontoon is mde in the shpe shown. The pontoon is designed rotting the grph of =,, out the -is, where nd re mesured in feet. Find the volume of the pontoon. ft To use the shell method in Emple, ou would hve to solve for in terms of in the eqution = nd then evlute n integrl tht requires u-sustitution. Sometimes, solving for is ver difficult (or even impossile). In such cses, ou must use verticl rectngle (of width Δ), 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. ft Shell Method Necessr Find the volume of the solid formed revolving the region ounded the grphs of = + +, =, nd = out the line =, s shown in Figure.. In the eqution = + +, ou cnnot esil solve for in terms of. (See the discussion t the end of Section.8.) 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. V = π p()h() d Appl shell method. = π ( )( + + ) d = π ( + + ) d = π [ + + ] = π ( + + ) = 9π Simplif. Integrte. Figure. (, ) p() = h() = + + Ais of revolution Aout. cuic feet Etr Emple Find the volume of the solid formed revolving the region ounded the grphs of = + +, =, nd = out the line =. π Teching Strtegies Point out tht p() is not lws. For instnce, in Emple, p() =. Eplin tht the vlue of p() is ffected the loction of the is of revolution. Lesson Closer Find the volume of the region ounded = +, =, nd = out the -is using () the wsher method nd () the shell method. () nd () 9π

28 . See mrgin. Chpter Applictions of Integrtion Assignment Guide Es. odd, 8, 9,,,,,. π d = π. π ( ) d = π. π d = 8π. π ( ) d = π. π ( ) d = π. π ( π d = ) 7. π ( ) d = π 8. π (9 ) d = 8π 9. π ( + ) d = 8π. π (8 ) d = 8π 7. π d = 8π. π ( ) d = π. π ( π e ) d = π ( e ).98 π. π ( sin ) d = π. π ( ) d = 8π. π ( )( + ) d = 8π 7. π[ d + ( ) d ] = π 8. π ( ) d = 8π 8 9. π d = 78π 7. π. π. π. Eercises See ClcCht.com for tutoril help nd worked-out solutions to odd-numered eercises. Finding the Volume of Solid 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.. See mrgin.. =. =. =. = +. =, =, =. =, =, = 7. =, = 8. = 9, = 9. =, =, =. =, = 8, =. =, =, =. = +, =. =, =, =, = π e. ={ sin, >, =, =, = π, = Finding the Volume of Solid 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.. See mrgin.. =. = ( π ) d = ( ) d = π ( + ) d = π 7. should e sutrcted from ; V = π ( )( ) d 7. = 9. =, =, = 8. =, =, =. + =, =, =. = +, =, = 8. + = 8 Finding the Volume of Solid 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 = 8π 9π. =, =, =, out the line =. =, =, out the line = π. =, =, out the line = π Find the Error Let A e the region enclosed the grphs of =, =, =, s shown. In Eercises 7, descrie nd correct the error in using the shell method to set up the given integrl which gives the volume V of the solid formed revolving the region out the line =. 7. See mrgin. 7. V = π ( ) d A 8. V = π ( + )( ) d 9. V = π ( )( + ) d. V = π ( )( ) d 8. should e sutrcted from ; V = π ( )( ) d 9. + should e sutrcted from ; V = π ( )( ) d. The limits of integrtion should e the -vlues nd ; V = π ( )( ) d

29 . See mrgin. Section. Volume: The Shell Method Choosing Method In Eercises nd, 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.). See mrgin.. ( ) =. = e Choosing Method In Eercises, use the disk method or the shell method to find the volumes of the solids generted revolving the region ounded the grphs of the equtions out the given lines.. See mrgin.. =, =, = () -is () -is (c) =. =, =, =, = () -is () -is (c) =. + =, =, = () -is () -is (c) =. + =, > (hpoccloid) () -is () -is Finding the Volume of Solid In Eercises 7, () 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. 7. See mrgin =, =, =, first qudrnt 8. =, =, = 9. = ( ) ( ), =, =, =. = + e, =, =, =. See mrgin. WRITING ABOUT CONCEPTS. Representtive Rectngles 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.. Descriing Clindricl Shells Consider the plne region ounded the grphs of = k, =, =, nd =, where k > nd >. Wht re the heights nd rdii of the clinders generted when this region is revolved out () the -is nd () the -is?. () Circle of rdius AB nd center A () Circulr clinder of rdius AB (c) Disk method: V = [g()] d. Shell method: V = π f () d. () Region ounded =, =, =, = () Revolved out the -is. () Region ounded =, =, = () Revolved out the -is WRITING ABOUT CONCEPTS (continued). Compring Integrls Give geometric rgument tht eplins wh the integrls hve equl vlues. () π ( ) d = π [ ( + )] d () π [ () ] d = π ( ) d. HOW DO YOU SEE IT? Use the grph to nswer the following. See mrgin. A = f() C B = g(). () Descrie the figure generted revolving segment AB out the -is. () Descrie the figure generted revolving segment BC out the -is. (c) Assume the curve in the figure cn e descried s = f () or = g(). A solid is generted revolving the region ounded the curve, =, nd = out the -is. Set up integrls to find the volume of this solid using the disk method nd the shell method. (Do not integrte.) Anlzing n Integrl In Eercises 8, the integrl represents the volume of solid of revolution. Identif () the plne region tht is revolved nd () the is of revolution. 8. See mrgin.. π d 7. π ( + ) d 8. π ( )e d 7. () Region ounded =, =, = () Revolved out = 8. () Region ounded = e, =, =, = () Revolved out = 9. Dimeter =.. Dimeter = π ( ) d 9. 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. See mrgin.. 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. See mrgin.. Shell method; It is much esier to put in terms of.. Shell method; It is much esier to integrte e thn ln( ).. () 8π 7. () 9π (c) 9π. () π. () π 7. () (). 8. (). () π () π ln () π. () π = ( / ) / (c) 9π (c) π.. (). 9. () 7 () 87.. ().. = = ( ) ( ) = + e / () 9.. () The rectngles would e verticl. () The rectngles would e horizontl.. () rdius = k; height = () rdius = ; height = k. () Both integrls ield the volume of the solid generted revolving the region ounded the grphs of =, =, nd = out the -is. () Both integrls ield the volume of the solid generted revolving the region ounded =, =, nd = out the -is. 7

30 . See mrgin. Chpter Applictions of Integrtion 7. () R (n) = n n + () lim n R (n) = n (c) V = π ( n+ n + ) R (n) = n n + (d) lim n R (n) = (e) As n, the grph pproches the line =.. Volume of Torus A torus is formed revolving the region ounded the circle + = out the line = (see figure). Find the volume of this doughnutshped solid. (Hint: The integrl d represents the re of semicircle.) π. Volume of Torus Repet Eercise for torus formed revolving the region ounded the circle + = r out the line = R, where r < R. π r R. Finding Volumes of Solids (i) () Use differentition to verif tht sin d = sin cos + C. Proof () Use the result of prt () to find the volume of the solid generted revolving ech plne region out the -is. V = π (ii) V = π = sin 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 πh (r h).. 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 is revolved out the -is is π. Wht is the volume when the region is revolved out the -is? Proof; V = π 7. Eplortion Consider the region ounded the grphs of = n, = n, nd = (see figure). See mrgin. n = n () Find the rtio R (n) of the re of the region to the re of the circumscried rectngle. () Find lim n R (n) nd compre the result with the re of the 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 n R (n) nd compre the result with the volume of the circumscried clinder. = sin = sin (e) Use the results of prts () nd (d) to mke conjecture out the shpe of the grph of = n ( ) s n.. Finding Volumes of Solids 8. Think Aout It Mtch ech integrl with the solid () Use differentition to verif tht whose volume it represents, nd give the dimensions of cos d = cos + sin + C. Proof ech solid. () Use the result of prt () to find the volume of the () Right circulr cone ii () Torus v (c) Sphere iii solid generted revolving ech plne region (d) Right circulr clinder i (e) Ellipsoid iv r r out the -is. (Hint: Begin pproimting (i) the points of intersection.) π h d (ii) π h ( r) d (i) V. (ii) = cos V.99 r = (iii) π r d. = ( ) = cos (iv) π. d r (v) π (R )( r ) d Proof r 9. Equl Volumes Let V nd V e the volumes of the solids tht result when the plne region ounded =, =, =, nd = c (where c > ) is revolved out the -is nd the -is, respectivel. Find the vlue of c for which V = V. c =

31 . See mrgin.. Volume of Segment of Proloid The region ounded = r, =, nd = is revolved out the -is to form proloid. A hole, centered long the is of revolution, is drilled through this solid. The hole hs rdius k, < k < r. Find the volume of the resulting ring () integrting with respect to nd () integrting with respect to. See mrgin.. Finding Volumes of Solids Consider the grph of = ( ) (see figure). Find the volumes of the solids tht re generted when the loop of this grph is revolved out () the -is, () the -is, nd (c) the line =. See mrgin. NASA = ( ) 7 Clculus AP Em Preprtion Questions. Multiple Choice The region shown in the figure is revolved out the -is, the -is, nd the line =. Which of the following orders the volumes of the resulting solids from lest to gretest? A (A) -is, =, -is (B) -is, -is, = (C) =, -is, -is (D) =, -is, -is = /. Multiple Choice Let A e the region ounded the grphs of =, =, nd =. Wht is the volume of the solid formed when A is revolved out the line =? C (A) π (B) π (C) π (D) π. Multiple Choice Let R e the region enclosed the grphs of = 8 ( ) nd =. Wht is the volume of the solid formed when R is revolved out the -is? D (A) π (B) 9π (C) 8π (D) π. Free Response Let T e the region ounded the grphs of = ( + ) nd =. () Find the re of the region..8 () Wht is the volume of the solid formed when T is revolved out the -is? V.888 (c) Wht is the volume of the solid formed when T is revolved out the -is? V = π Sturn Section. Volume: The Shell Method 7 The Olteness of Sturn Sturn is the most olte of the plnets in our solr sstem. Its equtoril rdius is,8 kilometers nd its polr rdius is, kilometers. The color-enhnced photogrph of Sturn ws tken Voger. In the photogrph, the olteness of Sturn is clerl visile. () Find the rtio of the volumes of the sphere nd the olte ellipsoid shown elow. () If plnet were sphericl nd hd the sme volume s Sturn, wht would its rdius e? Computer model of sphericl Sturn, whose equtoril rdius is equl to its polr rdius. The eqution of the cross section pssing through the pole is + =,8. Computer model of olte Sturn, whose equtoril rdius is greter thn its polr rdius. The eqution of the cross section pssing through the pole is,8 +, =. A worked-out solution to the Section Project cn e found in the Techer s Resource Mnul.. () nd () V = π (r k ). () π () 8π (c) 89π 7

32 8 Chpter Applictions of Integrtion Pul s Notes Section Overview An instructionl video from Pul, including teching strtegies for the section, is ville t LrsonClculusforAP.com.. 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 f 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 = < < <... < n = s shown in Figure.7. B letting Δ i = i i nd Δ i = i i, ou cn pproimte the length of the grph Essentil Question How cn ou use definite integrls to find the rc length of smooth curve nd the re of surfce of revolution? Tell students tht the will lern how to nswer this question comining limits, definite integrls, nd the Distnce Formul. Lesson Motivtor The length of n rc cn e estimted using stright line segments whose lengths re given the Distnce Formul. Comining this concept with definite integrls, students re le to find the ect rc length of smooth curve. This llows students to find the re of surfce of revolution using method similr to the shell method from Section.. s n ( i i ) + ( i i ) i= = n (Δ i ) + (Δ i ) i= = n Δ (Δ i) + i= ( i Δ i ) (Δ i ) = n + ( Δ i i= Δ i ) (Δ i ). This pproimtion ppers to ecome etter nd etter s Δ (n ). So, the length of the grph is s = lim Δ n + ( Δ i i= Δ i ) (Δ i ). Becuse f () eists for ech in ( i, i ), the Men Vlue Theorem gurntees the eistence of c i in ( i, i ) such tht f ( i ) f ( i ) = f (c i )( i i ) (, ) = Figure.7 (, ) (, ) = = s s = length of curve from to = n = f() ( n, n ) Teching Strtegies Note tht rc length nd surfces of revolution re topics to e covered in BC course onl. f ( i ) f ( i ) = f (c i i ) i Δ i = f (c Δ i ). i Becuse f is continuous on [, ], it follows tht + [ f ()] is lso continuous (nd therefore integrle) on [, ], which implies tht s = lim Δ n + [ f (c i )] (Δ i ) i= = + [ f ()] d where s is clled the rc length of f etween nd. 8

33 Section. Arc Length nd Surfces of Revolution 9 Definition of Arc Length Let the function = f () represent smooth curve on the intervl [, ]. The rc length of f etween nd is s = + [ f ()] d. Similrl, for smooth curve = 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. The Length of Line Segment Find the rc length from (, ) to (, ) on the grph of f () = m +, s shown in the figure. Becuse m = f () = it follows tht s = = + [ f ()] d + ( (, ) ) d (, ) f() = m + = ( ) + ( ) ( ) () ] = ( ) + ( ) ( ) ( ) = ( ) + ( ) Formul for rc length Integrte nd simplif. which is the formul for the distnce etween two points in the plne. Insight Finding the rc length of the grph of function over n intervl is tested on the AP Clculus BC Em. Technolog 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 rc lengths in Emples nd. Pul s Notes Common Errors When finding the rc length of curve, students often mistkenl write the integrl s + [f ()]. Stress to students tht the must tke the derivtive of the function first, s the correct integrl is + [f ()]. Etr Emple Use integrtion to find the rc length from (, ) to (, ) on the grph of f () = ( 7). Show tht ou get the sme nswer using the stndrd Distnce Formul for the length of line segment. ; ( ) + ( ) = 9 = Teching Strtegies Point out tht students cn still use the Distnce Formul to find the length of liner segment of curve. In fct, using the Distnce Formul is preferle in this cse, ecuse the process is much simpler. 9

34 Chpter Applictions of Integrtion Pul s Notes Teching Strtegies In generl, remind students to verif tht the derivtive of the given function is continuous t ll points on the given intervl efore finding the rc length of the curve. In cses where the derivtive is not continuous over the given intervl, it is est to chnge the vrile of integrtion or split the function into piecewise function. Etr Emple Find the rc length of the grph of = ( + 8) on the intervl [, ]. Etr Emple Find the rc length of the grph of 9 8 = on the intervl Finding Arc Length Find the rc length of the grph of = + on the intervl [, ], s shown in Figure.8. ) ields n rc length of Using d d = = ( s = + ( d d) d = + [ ( d )] = ( + + ) d = ( + = [ ] = ( + 7 ) =. ) d Simplif. Finding Arc Length Formul for rc length Integrte. Find the rc length of the grph of ( ) = on the intervl [, 8], s shown in Figure.9. On the intervl [, 8], dd = is not defined t. So, solve for in terms of : =±( ). Choosing the positive vlue of produces d d = ( ). The -intervl [, 8] corresponds to the -intervl [, ], nd the rc length is d s = + ( d c d) d = + [ ( ] ) d = 9 d = 9 d = 8[ (9 ) ] = 7 ( ) 9.7. Formul for rc length Simplif. Integrte. = + The rc length of the grph of on [, ] Figure.8 Alger Review For help on the lger in Emple, see Emple () in the Chpter Alger Review on pge A7. (, ) 7 8 The rc length of the grph of on [, 8] Figure.9 ( ) = (8, ) Alger Review For help on the lger in Emple, see Emple () in the Chpter Alger Review on pge A7.

35 Section. Arc Length nd Surfces of Revolution Finding Arc Length See LrsonClculusforAP.com for n interctive version of this tpe of emple. Find the rc length of the grph of = ln(cos ) from = to = π, s shown in Figure.. Using d d = sin cos = tn ields n rc length of s = + ( d d) d π = + tn d = = π π sec d sec d = [ ln sec + tn ] π = ln ( + ) ln.88. Formul for rc length Trigonometric identit Simplif. Integrte. Length of Cle An electric cle is hung etween two towers tht re feet prt, s shown in Figure.. The cle tkes the shpe of ctenr whose eqution is = 7(e + e ). Find the rc length of the cle etween the two towers. Becuse = (e e ), ou cn write The rc length of the grph of on [, π ] Figure. = ln(cos ) = 7(e / + e / ) Pul s Notes Etr Emple Find the rc length of the grph of = ln(csc ) from = π to = π. ln ( + ).7 Etr Emple An electric cle is hung etween two towers tht re feet prt, s shown. The cle tkes the shpe of ctenr whose eqution is = (e + e ). Find the rc length of the cle etween the two towers. = (e / + e / ) nd ( ) = (e7 + e 7 ) ( ) = (e e 7 ) = [ (e + e ) ]. Figure. Aout feet Therefore, the rc length of the cle is s = + ( ) d = (e + e ) d = 7 [ e e ] = (e e ) feet. Formul for rc length Integrte.

36 Chpter Applictions of Integrtion Are of Surfce of Revolution In Sections. nd., 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. Insight Finding the re of surfce of revolution is not tested on the AP Em. Definition of Surfce of Revolution When the grph of continuous function is revolved out line, the resulting surfce is surfce 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., 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 S = πrl Lterl surfce re of frustum where L r r Ais of revolution r = (r + r ). Averge rdius of frustum (In Eercise, ou re sked to verif the formul for S.) Consider function f tht hs continuous derivtive on the intervl [, ]. The grph of f is revolved out the -is to form surfce of revolution, s shown in Figure.. Let Δ e prtition of [, ], with suintervls of width Δ i. Then the line segment of length ΔL i = Δ i +Δ i Figure. = f() 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 Δ S i of the frustum is = i i = n Δ S i = πr i ΔL i = πf (d i ) Δ i +Δ i = πf (d i ) + ( Δ i Δ i ) Δ i. B the Men Vlue Theorem, point c i eists in ( i, i ) such tht f (c i ) = f ( i) f ( i ) i i = Δ i. Δ i So, ΔS i = π f (d i ) + [ f (c i )] Δ i, nd the totl surfce re cn e pproimted S π n f (d i ) + [ f (c i )] Δ i. i= It cn e shown tht the limit of the right side s Δ (n ) is S = π f () + [ f ()] d. In similr mnner, if the grph of f is revolved out the -is, then S is S = π + [ f ()] d. Figure. Ais of revolution

37 Section. Arc Length nd Surfces of Revolution In these two formuls for S, ou cn regrd the products π f () nd π s the circumferences of the circles trced point (, ) on the grph of f s it is revolved out the -is nd the -is (Figure.). 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 net 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 = π 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 = π 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 = π r() ds is function of. nd d S = π r() ds is function of. c where ds = + [ f ()] d nd ds = + [g ()] d, respectivel. 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.. The distnce etween the -is nd the grph of f is r() = f (), nd ecuse f () =, the surfce re is S = π r() + [ f ()] d = π + ( ) d = π ( )( + 9 ) d Simplif. = π 8[ ( + 9 ) ] = π 7 ( ) Formul for surfce re Integrte. Ais of revolution.. Figure. Ais of revolution Figure. r = f() r = f() = = f() (, f()) = f() (, f()) (, ) r() = f() Ais of revolution Pul s Notes Etr Emple Find the re of the surfce formed revolving the grph of f () = on the intervl [, ] out the -is, s shown. r() = Ais of revolution (, ) π. f() = Lesson Closer Find the perimeter of region R. Then find the surfce re of the solid formed revolving region R out the -is. R = + = / P = π(8 + ) S = 9.8

38 . See mrgin. Chpter Applictions of Integrtion Assignment Guide Es.,,, 7,,,,,, 8,,,,,,,. () nd () 7. () nd (). ( ).9. 7 (8 ) ln ( + ).7. ln( + ).7. ( e e ).7. = () () + d (c) Aout.7 8. () () + + d (c) Aout. 9. () () + d (c) Aout.7. (). Eercises See ClcCht.com for tutoril help nd worked-out solutions to odd-numered eercises. Finding Distnce Using Two Methods In Eercises nd, find the distnce etween the points using () the Distnce Formul nd () integrtion.. See mrgin.. (, ), (8, ). (, ), (7, ) Finding Arc Length In Eercises, find the rc length of the grph of the function over the indicted intervl.. = ( + ). = + = ( + ) / = +. = +. = +. See mrgin. = / + = / =, [, 8] 8. = 8 +, [, ] 9. = +, [, ]. = +, [, 7]. = ln(sin ), [ π, π ]. = ln(cos ), [, π ]. = (e + e ), [, ]. = e + e, [ln, ln ]. = ( + ),. = ( ), Finding Arc Length In Eercises 7, () sketch the grph of 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. 7. See mrgin. 7. =, () + ( + ) d (c) Aout. 7. () π () + cos d (c) Aout.8 8. = +, 9. =,. = +,. = sin, π. = cos, π π. = e,. = ln,. = rctn,. =, Approimtion In Eercises 7 nd 8, determine which vlue est pproimtes the length of the rc represented the integrl. (Mke our selection on the sis of sketch of the rc, not performing n clcultions.) 7. + [ d d( + )] d () () (c) (d) (e) π 8. + [ d d (tn ) ] d () () (c) (d) π (e) 9. Find the Error Descrie nd correct the error in finding the integrl tht gives the rc length s of the grph of = tn over the intervl [, ], where < < < π. The derivtive of should s = + ( d d) d e sec not sec ; s = = + (sec ) d + (sec ) d = + sec d + sec d =. Find the Error Descrie nd correct the error in finding the integrl tht gives the rc length s of the grph of = over the intervl [, ]. See mrgin. s = + ( ) d = = + ( d ) + 9. () d,. See Additionl Answers eginning on pge AA. π () + sin d π (c) Aout.8 e

39 . See mrgin. Section. Arc Length nd Surfces of Revolution. Length of Ctenr Electricl wires suspended etween two towers form ctenr (see figure) modeled the eqution = (e + e ),, where nd re mesured in meters. The towers re meters prt. Find the length of the suspended cle. 7.8 m. ROOF AREA A rn is feet long nd feet wide (see figure). 7. The integrl formul for the re of surfce of revolution is derived from the formul for the lterl surfce re of the frustum of right circulr cone. The formul is S = πrl, where r = (r + r ), which is the verge rdius of the frustum, nd L is the length of line segment on the frustum. The representtive element is πf (d i ) + ( Δ i Δ i ) Δ i. = (e / + e / ) ft A cross section of the roof is the inverted ctenr = (e + e ). Find the numer of squre feet of roofing on the rn. 7 ft. Astroid Find the totl length of the grph of the stroid + =. 8. Arc Length of Sector of Circle Find the rc length from (, ) clockwise to (, ) long the circle + =. Show tht the result is one-fourth the circumference of the circle. See mrgin. Finding the Are of Surfce of Revolution In Eercises, set up nd evlute the definite integrl for the re of the surfce generted revolving the curve out the -is.. See mrgin.. =,. =, 9 7. = +, 8. =, 9. =,. = 9, istockphoto.com/wwing Finding the Are of Surfce of Revolution In Eercises, set up nd evlute the definite integrl for the re of the surfce generted revolving the curve out the -is.. See mrgin.. = +, 8. = 9,. =,. = +, 7. See mrgin. WRITING ABOUT CONCEPTS. Rectifile Curve Define rectifile curve.. Preclculus nd Clculus Wht preclculus formul nd representtive element re used to develop the integrtion formul for rc length? 7. Preclculus nd Clculus Wht preclculus formul nd representtive element re used to develop the integrtion formul for the re of surfce of revolution? 8. HOW DO YOU SEE IT? The grphs of the functions f nd f on the intervl [, ] re shown in the figure. The grph of ech function is revolved out the -is. Which surfce of revolution hs the greter surfce re? Eplin. The surfce of revolution given f will e lrger ecuse r() is lrger for f. 9. () (),,, (c) s.7; s.79; s.9; s. f 9. Think Aout It The figure shows the grphs of the functions =, =, =, nd = 8 on the intervl [, ]. To print n enlrged cop of the grph, go to MthGrphs.com. See mrgin. f () Lel the functions. () List the functions in order of incresing rc length. (c) Verif our nswer in prt () using the integrtion cpilities of grphing utilit.. 7.8; [π()] 7.8 = s. π + d = π (8 8 ) π + d 7. π ( + )( = 8π ( ) ) d = 7π π d = 7 π π. π 8. π d = 8π. 9 9 d + 9 d = π 7. = π ( ) π + d = π (7 ) 7.9. π + d = π ( 8).8. π d = π A rectifile curve is curve with finite rc length.. The preclculus formul is the distnce formul etween two points. The representtive element is (Δ i ) + (Δ i ) = + ( Δ i Δ i ) Δ i.

40 . See mrgin.. () Answers will vr. Smple nswer: 7. in. () Answers will vr. Smple nswer: 8. in. (c) r = (d) 79. in. ; 79. in.. See elow. 7. () π ( ) () π + d (c) lim V = lim [ π ( )] = π (d) Becuse + > = > on [, ], ou hve + d > d = [ ln ] = ln nd lim ln. So, lim π + d =. Chpter Applictions of Integrtion. Verifing Formul () Given circulr sector with rdius L nd centrl ngle θ (see figure), show tht the re of the sector is given S = L θ. () 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 = πrl, 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.) Proof L Proof Figure for () Figure for () (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 = π(r + r )L. (Note: This formul ws used to develop the integrl for finding the surfce re of surfce of revolution.) Proof L r r Ais of revolution. Lterl Surfce Are of Cone A right circulr cone is generted revolving the region ounded =, =, nd = out the -is. Find the lterl surfce re of the cone. π. Lterl Surfce Are of Cone 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 = πr r + h. Proof. Using Sphere Find the re of the zone of sphere formed revolving the grph of = 9,, out the -is. π( ).. Using Sphere Find the re of the zone of sphere formed revolving the grph of = r,, out the -is. Assume tht < r. πrh r L See mrgin.. 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 verticl distnce in inches from the se. 9 8 C () 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(π). 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 ().. Modeling Dt Propert ounded two perpendiculr rods nd strem is shown in the figure. All distnces re mesured in feet. See mrgin. (, ) (, 9) (, ) (,) (, ) (, 7) (, ) (, ) (, 9) () 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. 7. Volume nd Surfce Are Let R e the region ounded =, the -is, =, nd =, where >. Let D e the solid formed when R is revolved out the -is. See mrgin. () 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.. () f () = () Answers will vr. Smple nswer:. cres (c) Answers will vr. Smple nswer: 79.9 ft

41 . See mrgin. 8. Think Aout It Consider the eqution 9 + =. () Use grphing utilit to grph the eqution. () Set up the definite integrl for finding the firstqudrnt 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? Eplin. (You will lern how to evlute this tpe of integrl in Section 7.8.) Approimting Arc Length or Surfce Are In Eercises 9 nd, set up the definite integrl for finding the indicted rc length or surfce re. Do not integrte. (You will lern how to evlute this tpe of integrl in Section 7.8.) 9. See mrgin. 9. 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. = ( / / + ) Figure for 9 Figure for Section. Arc Length nd Surfces of Revolution 7 See mrgin.. SUSPENSION BRIDGE = / /. 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. inch thick.). 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 C = w + (h w ) d. Proof h The Humer Bridge, locted in the United Kingdom nd opened in 98, hs min spn of out meters. Ech of its towers hs height of out meters. Use these dimensions, the integrl in Eercise, nd the integrtion cpilities of grphing utilit to pproimte the length of prolic cle long the min spn.. m Clculus AP Em Preprtion Questions. Multiple Choice Which of the following integrls gives the rc length of the grph of = cos etween nd, where < <? B (A) + cos d (B) + sin d (C) + sin d (D) cos + sin d. Multiple Choice The length of curve from = to = is given + d. If the curve contins the point (, 9), which of the following could e n eqution for this curve? (A) = + 7 (B) = + A (C) = (D) = + 8. Multiple Choice The figure shows the grphs of f () = nd g() = +. Wht is the length of the curve = f () etween the points of intersection shown in the figure? C (A) 8 (B) 8. (C) 8. (D) () + = 9 () d (c) You cnnot evlute this definite integrl ecuse the integrnd is not defined t =. Also, the integrnd does not hve n elementr ntiderivtive. 9. Fleeing oject: unit Pursuer: + d. See elow. w istockphoto.com/pulvinten. π ( ) ( + 9 ) d 7

42 . See mrgin. 8 Chpter Applictions of Integrtion Assignment Guide Es.,, 8 even, 9,,,, 7, 8,, 7,,. (sin π + ) d.9. (e + ) d.. ( + 8) d. ( + ) d (, ) (, ), (, ), (, ) ( ),,, Review Eercises See ClcCht.com for tutoril help nd worked-out solutions to odd-numered eercises.. See mrgin. Writing Definite Integrl In Eercises, set up the definite integrl tht gives the re of the region.. = sin π. = + = = e. = + 8. = ( ) + = = Finding the Are of Region In Eercises, sketch the region ounded the grphs of the equtions nd find the re of the region.. See mrgin.. =, =, =, =. =, =, = 7. =, =, =, = + 8. =, =, = 9. =, =. = +, = +. = e, = e, =. = csc, =, π π. = sin, = cos, π π. = cos, =, π 7π Finding the Are of Region In Eercises 8, use grphing utilit to grph the region ounded the grphs of the equtions, nd use the integrtion cpilities of the grphing utilit to find the re of the region. 8. See mrgin.. = 8 +, = + 8. = +, =, = 7. + =, =, = 8. =, = 9. Numericl Integrtion Estimte the surfce re of the pond using the Trpezoidl Rule. 99 ft ft 8 ft 7 ft ft 8 ft 7 ft ft 8 ft. Revenue The models R =. +.t +.t nd R = 8. +.t give the revenue (in illions of dollrs) for lrge corportion. Both models re estimtes of the revenues from through, with t = corresponding to. Which model projects the greter revenue? How much more totl revenue does tht model project over the si-er period? See mrgin.. Using Tngent Line Find the re of the region ounded the grph of f () = nd the tngent line to the grph t the point (, ). See mrgin.. Using Tngent Line Find the re of the region ounded the grph of g() = ( + ) nd the tngent line to the grph t the point (, ). See mrgin. Finding the Volume of Solid In Eercises, use the disk method or the shell method to find the volumes of the solids generted revolving the region ounded the grphs of the equtions out the given line(s).. =, =, = See mrgin. () -is () -is (c) = (d) =. =, =, = See mrgin. () -is () = (c) -is (d) =. =, =, =, = + π revolved out the -is 8. π (, ) (, ) (, ) 9. (, ) (, ) (, ). 9 (, ) (, ). The second model projects the greter revenue; $7.7 illion. 7. π.. () 9π () 8π (c) 9π (d) π. () 8π () 8π (c) π (d) 7π 8 8. See Additionl Answers eginning on pge AA.

43 . See mrgin.. = +, =, =, = π revolved out the -is 7. =, =, =, = revolved out the -is π ln = e, =, =, = π revolved out the -is ( e ).8 9. =, = π revolved out the -is. = sec, =, =, = π π revolved out the -is. Depth of Gsoline in Tnk A gsoline tnk is n olte spheroid generted revolving the region ounded the grph of + 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..98 ft. Using Cross Sections Find the volume of the solid whose se is ounded the circle + = 9 nd the cross sections perpendiculr to the -is re equilterl tringles. Finding Distnce Using Two Methods In Eercises nd, find the distnce etween the points using () the Distnce Formul nd () integrtion.. (, ), (, ). (, ), (9, 8) () nd () () nd () 7 Finding Arc Length In Eercises nd, find the rc length of the grph of the function over the indicted intervl. Review Eercises 9 8. Approimtion Determine which vlue est pproimtes the length of the rc represented the integrl + [ d d( + )] d. (Mke our selection on the sis of sketch of the rc nd not performing n clcultions.) c () () (c) (d) (e) 9. Arc Length of Sector of Circle Find the rc length from (, ) clockwise to (, ) long the circle + = 9. See mrgin.. Arc Length of Sector of Circle Find the rc length from (, ) clockwise to (, 7) long the circle + =. See mrgin.. Surfce Are Use integrtion to find the lterl surfce re of right circulr cone of height nd rdius. π. Surfce Are The region ounded the grphs of =, =, =, nd = 8 is revolved out the -is. Find the surfce re of the solid generted. See mrgin. Approimting Surfce Are In Eercises nd, set up the definite integrl for finding the indicted surfce re. Do not integrte. (You will lern how to evlute this tpe of integrl in Section 7.8.). Astroid Find the re of the surfce formed revolving the portion in the first qudrnt of the grph of + =, 8, 8 out the -is. π ( ) d ( + ).7 9. rcsin.89. rcsin + π.8. π See. f () =. = + mrgin.. Using Loop Consider the grph of = ( ) = / = + shown in the figure. Find the re of the surfce formed when the loop of this grph is revolved out the -is. = ( ) π 7. Length of Ctenr A wire hung etween two poles forms ctenr modeled the eqution = (e + e ), where nd re mesured in feet. Find the length of the wire etween the two poles. (e e ). ft 9

44 . See mrgin. Chpter Applictions of Integrtion AP Em Prctice Questions for Chpter Wht You Need to Know... The shell method is not required on the AP Em, ut some free-response questions m e solvle the shell method in ddition to the disk method. The shell method is prticulrl dvntgeous when it is difficult to epress one vrile in terms of the other nd when more thn one integrl is required to find volume. On the AP Clculus BC Em, ou m need to use the concept of rc length to find the perimeter of given region. When limits of integrtion re irrtionl numers, ou cn ssign vriles to represent ech limit of integrtion. You cn then write the correct integrl epression using the vriles, insted of writing out ech integrl in its entiret. This will help ou sve time. Some questions m just sk ou to write, nd not necessril evlute, the correct integrl ou cn use to solve the prolem. Prctice Questions Section, Prt A, Multiple Choice, No Technolog. Wht is the re of the region ounded the -is, the line = e, nd the grph of the function = e? A (A) (B) e e (C) e (D) 8 e. Wht is the re enclosed the curves = nd = +? C (A) (B) (C) (D) In Eercises nd, use the figure shown elow. Let R e the region ounded the grphs of = cos, = sin, nd the -is.. Which epression represents the re of R? (A) ( ) (B) (C) ( ) (D) A. The horizontl line = splits the region R into two prts. Wht is the re of the prt of R tht is elow this horizontl line? B (A) π (B) π + (C) π + (D) π +. Wht is the re of the region ounded the curves = nd = +? D (A) 9 (B) 8 (C) (D). Which of the following integrls gives the length of the grph of = ln(sec ) from = to = π? B (A) π sec d (B) π sec d (C) π sec tn d (D) π + cos d 7. Which of the following integrls gives the length of the grph of = e. etween nd? A (A) + e d (B) + e d (C) + e. d (D) + e d 8. Wht is the rc length of the grph of = from = to = 8? B (A) (C) (B) 8 (D) 8

45 . See mrgin. AP Em Prctice Questions for Chpter Section, Prt B, Multiple Choice, Technolog Permitted 9. The se of solid is the region in the first qudrnt ounded ove the line =, elow = sin, nd to the right the line =. For this solid, ech cross section perpendiculr to the -is is squre. Wht is the volume of the solid? D (A).9 (B) (C).8 (D) Section, Prt A, Free Response, Technolog Permitted. Consider the region ounded the -is, =, nd = +. () Write, ut do not evlute, n integrl eqution tht will find the vlue of k so tht = k divides the region into two prts of equl re. See mrgin. () Find the length of the curve = + on the intervl [, ].. (c) The region is the se of solid. For this solid, the cross sections perpendiculr to the -is re rectngles with height of times tht of its width. Find the volume of this solid..9. Let R e the region ounded the grphs of = ln nd =. () Find the re of R..7 () Find the volume of the solid generted when R is rotted out the horizontl line = (c) Write, ut do not evlute, n epression involving one or more integrls tht cn e used to find the volume of the solid generted when R is revolved out the -is. See mrgin.. Let R e the region ounded the grphs of = nd =. () Find the re of R..77 () Find the volume of the solid generted when R is rotted out the verticl line =.. (c) Write, ut do not evlute, n epression involving one or more integrls to find the volume of the solid generted when R is rotted out the horizontl line =. See mrgin. Section, Prt B, Free Response, No Technolog. A region in the -plne is ounded = +, = +, =, nd =. See mrgin. () Sketch the ounded region. Lel ech oundr curve nd shde the ounded region. () Find the re of the ounded region. Show the work tht leds to our nswer.. The region shown elow is ounded f () =, =, =, nd =. = () Find the volume of the solid formed rotting the region out the -is. π () Find the volume of the solid formed rotting the region out the -is. See mrgin. (c) Write n epression tht gives the volume of the solid formed rotting the region out the line =. See mrgin. (d) The region shown is the se of solid. For this solid, ech cross section perpendiculr to the -is is n equilterl tringle. Find the volume of this solid. See mrgin.. Consider the region R ounded the grphs of =, = 8, nd the -is. The region S is ounded =, =, nd the -is. () Find the re of R. () Find the volume of the solid formed rotting R out the -is. See mrgin. (c) The region S is the se of solid. For this solid, ech cross section perpendiculr to the -is is semicircle with dimeters etending from = to the -is. Find the volume of this solid. See mrgin.. Consider the region T ounded the grphs of =, =, nd =. See mrgin. () Find the re of T. () Find the volume of the solid formed rotting T out the horizontl line =. (c) Write, ut do not evlute, n epression involving one or more integrls tht gives the perimeter of T.. () ( ) d. (c).8788 π [( (c) See elow.. () +k = ( ) d ) (e ) ] d = = + = + () See elow. =. () π.7 (c) π ( ) d (d). () 9π.8 (c) π () () 7π.7 (c) P = d.888. (c) π. () A = [(.9 [( + ) ( + ) ] d + π + ) ( )] d = ( ) d = [ [( + ) ( ) ] d + ] =

46 . See mrgin. Chpter Applictions of Integrtion Smple Ruric Eercise... Totl Points Points Totl possile points for eplining how to find the re without clculus for finding the re for nswering no for eplining wh not Totl possile points for eplining how to find the re with clculus for finding the re for nswering es or no for eplining wh or wh not Totl possile points for eplining how to find the volume of the concrete needed for finding the volume points. Answers will vr. Smple nswer: The re of cross section cn e pproimted using geometric formuls. The cross section of the dm on 7 is nerl tringle, so the re cn e estimted using the re of tringle formul. The cross section of the dm on 9 is trpezoid, so the re is found using the re of trpezoid formul;,87. ft ; No. The re of the cross section is not ect ecuse the cross section of the dm on 7 is not tringle ecuse the portion of the grph from ( 7, ) to (, ) is not line.. To clculte the re of the cross section, integrte the piecewise function f on ech intervl nd dd the results;,.7 ft ; If the model of the cross section is ect, then the re is ect. If the model is n pproimtion of the cross section, then the re is n pproimtion. Performnce Tsk Constructing n Arch Dm One design used in dm construction is the rch dm. This design curves towrd the wter it contins, nd is usull uilt in nrrow cnons. The force of the wter presses the edges of the dm ginst the wlls of the cnon, so tht the nturl rock helps support the structure. Shown t the right is fmous use of this design, the Hoover Dm in the Blck Cnon of the Colordo River. Eercises In Eercises, use the following informtion. A cross section of n rch dm cn e modeled s shown in the figure t the right. The model for this cross section is f () ={ , 89, , 7 < <. 9 To form the rch dm, this cross section is swung through n rc, rotting it out the -is (see figure elow). The cross section is rotted nd the is of rottion is feet.. See mrgin.. To find the volume of concrete needed to uild the dm, multipl the shell method formul for verticl is of revolution ecuse the cross section is rotted onl. Then integrte the new formul;. 7 ft. Clculting Are without Clculus Eplin how ou would clculte the re of cross section of the dm without using clculus. Use our eplntion to clculte the re of the cross section. Is the re ou found ect? Wh or wh not?. Clculting Are with Clculus Eplin how ou would clculte the re of cross section of the dm with clculus. Use our eplntion to clculte the re of the cross section. Is the re ou found ect? Wh or wh not?. Clculting Volume Eplin how ou would clculte the volume of concrete needed to uild the dm. Use our eplntion to clculte this volume. istockphoto.com/strtol (, 89) (, 89) (, ) (7, ) (9, ) 8 8 (, ) A cross section of n rch dm