Arc Length and Surfaces of Revolution. Find the arc length of a smooth curve. Find the area of a surface of revolution. <...

Transcription

1 76 CHAPTER 7 Applictions of Integrtion The Dutch mthemticin Christin Hugens, who invented the pendulum clock, nd Jmes Gregor (6 675), Scottish mthemticin, oth mde erl contriutions to the prolem of finding the length of rectifile curve MthBio (, ) Section 7 (, ) (, ) = = Figure 77 CHRISTIAN HUYGENS (69 695) Histor = s s = length of curve from to = n = f() ( n, n ) Arc Length nd Surfces of Revolution Find the rc length of smooth curve Find the re of surfce of revolution Arc Length In this section, definite integrls re used to find the rc lengths of curves nd the res of surfces of revolution In either cse, n rc ( segment of curve) is pproimted stright line segments whose lengths re given the fmilir Distnce Formul d A rectifile curve is one tht hs finite rc length You will see tht sufficient condition for the grph of function f to e rectifile etween, f nd, f is tht e continuous on, Such function is continuousl differentile on,, nd its grph on the intervl, is smooth curve Consider function f tht is continuousl differentile on the intervl, You cn pproimte the grph of f n line segments whose endpoints re determined the prtition f < < < < n s shown in Figure 77 B letting i i i nd i i i, ou cn pproimte the length of the grph s n i i i i i n i i i n i i i i i n i i i i This pproimtion ppers to ecome etter nd etter s n So, the length of the grph is s lim n i i i i Becuse f eists for ech in i, i, the Men Vlue Theorem gurntees the eistence of in i, i such tht f i f i fc i i i Becuse is continuous on,, it follows tht f is lso continuous (nd therefore integrle) on,, which implies tht f s lim n c i i i fc i i fc i i f d where s is clled the rc length of f etween nd

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

3 7 CHAPTER 7 Applictions of Integrtion = + 6 EXAMPLE Finding Arc Length Find the rc length of the grph of 6 on the intervl,, s shown in Figure 79 The rc length of the grph of on Figure 79 Editle Grph, FOR FURTHER INFORMATION To see how rc length cn e used to define trigonometric functions, see the rticle Trigonometr Requires Clculus, Not Vice Vers Yves Nievergelt in UMAP Modules Solution Using d d 6 ields n rc length of s d Tr It d d 6 6 Eplortion A d d 7 d Formul for rc length Simplif Integrte EXAMPLE Finding Arc Length (, 5) 5 ( ) = (, ) The rc length of the grph of on, Figure 7 Editle Grph Find the rc length of the grph of on the intervl,, s shown in Figure 7 Solution Begin solving for in terms of : ± Choosing the positive vlue of produces d d The -intervl, corresponds to the -intervl, 5, nd the rc length is Formul for rc length d 5 d d s d c d 9 5 d d Simplif Integrte Tr It Eplortion A

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

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

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

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

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

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

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

11 6 CHAPTER 7 Applictions of Integrtion 6 (, 5) (5, 9) Figure for 5 (, 9) 59 Let R e the region ounded, the -is,, nd, where > Let D e the solid formed when R is revolved out the -is () Find the volume V of D () Write the surfce re S s n integrl (c) Show tht V pproches finite limit s (d) Show tht S s 6 () Given circulr sector with rdius L nd centrl ngle (see figure), show tht the re of the sector is given S L (5, ) (,5) (5, 6) (, 75) (5, 5) (, ) () B joining the stright line edges of the sector in prt (), right circulr cone is formed (see figure) nd the lterl surfce re of the cone is the sme s the re of the sector Show tht the re is S rl, where r is the rdius of the se of the cone (Hint: The rc length of the sector equls the circumference of the se of the cone) θ L 6 r L 6 Individul Project Select solid of revolution from everd life Mesure the rdius of the solid t minimum of seven points long its is Use the dt to pproimte the volume of the solid nd the surfce re of the lterl sides of the solid 6 Writing Red the rticle Arc Length, Are nd the Arcsine Function Andrew M Rockett in Mthemtics Mgzine Then write prgrph eplining how the rcsine function cn e defined in terms of n rc length MthArticle 6 Astroid Find the re of the surfce formed revolving the portion in the first qudrnt of the grph of, out the -is Figure for 6 Figure for 6 Rottle Grph 6 Consider the grph of (see figure) Find the re of the surfce formed when the loop of this grph is revolved round the -is 65 Suspension Bridge A cle for suspension ridge hs the shpe of prol with eqution k Let h represent the height of the cle from its lowest point to its highest point nd let w represent the totl spn of the ridge (see figure) Show tht the length C of the cle is given w h C w d 5 6 = ( ) Figure for 6() Figure for 6() (c) Use the result of prt () to verif tht the formul for the lterl surfce re of the frustum of cone with slnt height L nd rdii r nd r (see figure) is S r r L (Note: This formul ws used to develop the integrl for finding the surfce re of surfce of revolution) L r r Rottle Grph w 66 Suspension Bridge The Humer Bridge, locted in the United Kingdom nd opened in 9, hs min spn of out meters Ech of its towers hs height of out 55 meters Use these dimensions, the integrl in Eercise 65, nd the integrtion cpilities of grphing utilit to pproimte the length of prolic cle long the min spn h Rottle Grph Ais of revolution Putnm Em Chllenge 67 Find the length of the curve from the origin to the point where the tngent mkes n ngle of 5 with the -is This prolem ws composed the Committee on the Putnm Prize Competition The Mthemticl Assocition of Americ All rights reserved