Area of a Region Between Two Curves

Transcription

1 6 CHAPTER 7 Applictions o Intertion Section 7 Are o Reion Between Two Curves Find the re o reion etween two curves usin intertion Find the re o reion etween intersectin curves usin intertion Descrie intertion s n ccumultion process Are o Reion Between Two Curves Reion etween two curves With ew modiictions ou cn etend the ppliction o deinite interls rom the re o reion under curve to the re o reion etween two curves Consider two unctions nd tht re continuous on the intervl, I, s in Fiure 7, the rphs o oth nd lie ove the -is, nd the rph o lies elow the rph o, ou cn eometricll interpret the re o the reion etween the rphs s the re o the reion under the rph o sutrcted rom the re o the reion under the rph o, s shown in Fiure 7 = = Fiure 7 Are o reion etween nd d Fiure 7 Are o reion under d Are o reion under d Animtion ( i ) Fiure 7 Representtive rectnle Heiht: ( i ) ( i ) Width: i ( i ) To veri the resonleness o the result shown in Fiure 7, ou cn prtition the intervl, into n suintervls, ech o width Then, s shown in Fiure 7, sketch representtive rectnle o width nd heiht i i, where i is in the ith intervl The re o this representtive rectnle is A i heihtwidth i i B ddin the res o the n rectnles nd tkin the limit s n, ou otin lim n n i i i Becuse nd re continuous on,, is lso continuous on, nd the limit eists So, the re o the iven reion is Are lim n n i i i d

2 SECTION 7 Are o Reion Between Two Curves 7 Are o Reion Between Two Curves I nd re continuous on, nd or ll in,, then the re o the reion ounded the rphs o nd nd the verticl lines nd is A d In Fiure 7, the rphs o nd re shown ove the -is This, however, is not necessr The sme internd cn e used s lon s nd re continuous nd or ll in the intervl, This result is summrized rphicll in Fiure 7 () () (, ()) () () (, ()) (, ()) (, ()) NOTE The heiht o representtive rectnle is rerdless o the reltive position o the -is, s shown in Fiure 7 Fiure 7 Representtive rectnles re used throuhout this chpter in vrious pplictions o intertion A verticl rectnle o width implies intertion with respect to, wheres horizontl rectnle o width implies intertion with respect to EXAMPLE Findin the Are o Reion Between Two Curves () = + (, ()) (, ()) () = Reion ounded the rph o, the rph o,, nd Fiure 7 Find the re o the reion ounded the rphs o,,, nd Solution Let nd Then or ll in,, s shown in Fiure 7 So, the re o the representtive rectnle is A nd the re o the reion is A d d 7 6 Editle Grph Tr It Eplortion A Eplortion B

3 8 CHAPTER 7 Applictions o Intertion Are o Reion Between Intersectin Curves In Emple, the rphs o nd do not intersect, nd the vlues o nd re iven eplicitl A more common prolem involves the re o reion ounded two intersectin rphs, where the vlues o nd must e clculted EXAMPLE A Reion Lin Between Two Intersectin Grphs Find the re o the reion ounded the rphs o nd (, ()) () = () = (, ()) Reion ounded the rph o nd the rph o Fiure 76 Solution In Fiure 76, notice tht the rphs o nd hve two points o intersection To ind the -coordintes o these points, set nd equl to ech other nd solve or Set equl to Write in enerl orm Fctor or Solve or So, nd Becuse or ll in the intervl,, the representtive rectnle hs n re o A nd the re o the reion is A d 9 Editle Grph Tr It EXAMPLE Eplortion A A Reion Lin Between Two Intersectin Grphs The sine nd cosine curves intersect ininitel mn times, oundin reions o equl res, s shown in Fiure 77 Find the re o one o these reions π (, ()) (, ()) π π () = cos () = sin One o the reions ounded the rphs o the sine nd cosine unctions Fiure 77 Solution sin cos sin cos Set equl to Divide ech side cos tn or, Trionometric identit Solve or So, nd Becuse sin cos or ll in the intervl,, the re o the reion is A sin cos d cos sin Editle Grph Tr It Eplortion A

4 SECTION 7 Are o Reion Between Two Curves 9 I two curves intersect t more thn two points, then to ind the re o the reion etween the curves, ou must ind ll points o intersection nd check to see which curve is ove the other in ech intervl determined these points EXAMPLE Curves Tht Intersect t More Thn Two Points Find the re o the reion etween the rphs o nd () () () () 6 (, ) (, ) 6 (, 8) 8 () = + () = On,,, nd on,, Fiure 78 Editle Grph Solution Bein settin nd equl to ech other nd solvin or This ields the -vlues t ech point o intersection o the two rphs,, 6 6 Set equl to Write in enerl orm Fctor Solve or So, the two rphs intersect when,, nd In Fiure 78, notice tht on the intervl, However, the two rphs switch t the oriin, nd on the intervl, So, ou need two interls one or the intervl, nd one or the intervl, A d d d d Tr It Eplortion A Open Eplortion NOTE In Emple, notice tht ou otin n incorrect result i ou interte rom to Such intertion produces d d I the rph o unction o is oundr o reion, it is oten convenient to use representtive rectnles tht re horizontl nd ind the re intertin with respect to In enerl, to determine the re etween two curves, ou cn use A top curve ottom curve d Verticl rectnles in vrile A riht curve let curve d Horizontl rectnles in vrile where, nd, re either djcent points o intersection o the two curves involved or points on the speciied oundr lines Technolo

5 CHAPTER 7 Applictions o Intertion EXAMPLE Horizontl Representtive Rectnles Find the re o the reion ounded the rphs o nd Solution Consider nd These two curves intersect when nd, s shown in Fiure 79 Becuse on this intervl, ou hve A So, the re is A d d 8 9 Tr It Eplortion A () = + (, ) = (, ) = (, ) () = Horizontl rectnles (intertion with respect to ) Fiure 79 Editle Grph (, ) = Verticl rectnles (intertion with respect to ) Fiure 7 In Emple, notice tht intertin with respect to ou need onl one interl I ou hd interted with respect to, ou would hve needed two interls ecuse the upper oundr would hve chned t, s shown in Fiure 7 A d d 6 d d 9

6 SECTION 7 Are o Reion Between Two Curves Intertion s n Accumultion Process In this section, the intertion ormul or the re etween two curves ws developed usin rectnle s the representtive element For ech new ppliction in the reminin sections o this chpter, n pproprite representtive element will e constructed usin preclculus ormuls ou lred know Ech intertion ormul will then e otined summin or ccumultin these representtive elements Known preclculus ormul Representtive element New intertion ormul For emple, in this section the re ormul ws developed s ollows A heihtwidth A A d EXAMPLE 6 Descriin Intertion s n Accumultion Process Find the re o the reion ounded the rph o nd the -is Descrie the intertion s n ccumultion process Solution The re o the reion is iven A d You cn think o the intertion s n ccumultion o the res o the rectnles ormed s the representtive rectnle slides rom to, s shown in Fiure 7 A d A d A d 6 A d 9 Fiure 7 A d Tr It Eplortion A

7 CHAPTER 7 Applictions o Intertion Eercises or Section 7 The smol Click on Click on indictes n eercise in which ou re instructed to use rphin technolo or smolic computer ler sstem to view the complete solution o the eercise to print n enlred cop o the rph In Eercises 6, set up the deinite interl tht ives the re o the reion In Eercises nd, ind the re o the reion intertin () with respect to nd () with respect to Think Aout It In Eercises nd 6, determine which vlue est pproimtes the re o the reion ounded the rphs o nd (Mke our selection on the sis o sketch o the reion nd not perormin n clcultions) In Eercises 7, the internd o the deinite interl is dierence o two unctions Sketch the rph o ech unction nd shde the reion whose re is represented the interl 7 8 d d sec d d d sec cos d, () () (c) (d) (e) 8 6, () () 6 (c) (d) (e) In Eercises 7, sketch the reion ounded the rphs o the leric unctions nd ind the re o the reion 7,,, 8 8 8,,, 8 9,,,,,,, 6, 7, 8, 9,,,,,,,, 6,,,,,

8 SECTION 7 Are o Reion Between Two Curves In Eercises, () use rphin utilit to rph the reion ounded the rphs o the equtions, () ind the re o the reion, nd (c) use the intertion cpilities o the rphin utilit to veri our results,,,, 6, 7, 8, 9, 6,,,,,, In Eercises 8, sketch the reion ounded the rphs o the unctions, nd ind the re o the reion sin, tn, sin, cos, 6 cos, cos, 6 In Eercises 9, () use rphin utilit to rph the reion ounded the rphs o the equtions, () ind the re o the reion, nd (c) use the intertion cpilities o the rphin utilit to veri our results In Eercises 6, () use rphin utilit to rph the reion ounded the rphs o the equtions, () eplin wh the re o the reion is diicult to ind hnd, nd (c) use the intertion cpilities o the rphin utilit to pproimte the re to our deciml plces sec tn 7 e,, 8, 9 sin sin,, sin cos,, < e,, ln,,,, e,,, 6,, cos,, In Eercises 7 6, ind the ccumultion unction F Then evlute F t ech vlue o the independent vrile nd rphicll show the re iven ech vlue o F 7 F t dt () F () F (c) F6 8 F t dt () F () F (c) F6 9 F cos () F () F (c) F d 6 F e d () F () F (c) F In Eercises 6 6, use intertion to ind the re o the iure hvin the iven vertices 6,,, 6, 6, 6,,,,, c 6 6,,,,,,,,,,,,,, 6 Numericl Intertion Estimte the surce re o the ol reen usin () the Trpezoidl Rule nd () Simpson s Rule t t 66 Numericl Intertion Estimte the surce re o the oil spill usin () the Trpezoidl Rule nd () Simpson s Rule mi t mi t mi t mi t In Eercises 67 7, set up nd evlute the deinite interl tht ives the re o the reion ounded the rph o the unction nd the tnent line to the rph t the iven point mi t mi t 6 t mi mi 6 t 67,, 68, 69 7,, Writin Aout Concepts,,, 7 The rphs o nd intersect t three points However, the re etween the curves cn e ound sinle interl Eplin wh this is so, nd write n interl or this re

9 CHAPTER 7 Applictions o Intertion Writin Aout Concepts (continued) 7 The re o the reion ounded the rphs o nd cnnot e ound the sinle interl d Eplin wh this is so Use smmetr to write sinle interl tht does represent the re 7 A collee rdute hs two jo oers The strtin slr or ech is $,, nd ter 8 ers o service ech will p $, The slr increse or ech oer is shown in the iure From strictl monetr viewpoint, which is the etter oer? Eplin Slr (in dollrs) 6,,,,,, S Fiure or 7 Fiure or 7 7 A stte leislture is detin two proposls or elimintin the nnul udet deicits the er The rte o decrese o the deicits or ech proposl is shown in the iure From the viewpoint o minimizin the cumultive stte deicit, which is the etter proposl? Eplin In Eercises 7 nd 76, ind such tht the line divides the reion ounded the rphs o the two equtions into two reions o equl re 7 9, 76 In Eercises 77 nd 78, ind such tht the line divides the reion ounded the rphs o the equtions into two reions o equl re 77,, 78, In Eercises 79 nd 8, evlute the limit nd sketch the rph o the reion whose re is represented the limit 79 lim i i, where i in nd n n i 8 lim i, where i in nd n n i Oer Oer 6 8 Yer t Deicit (in illions o dollrs) 9, Revenue In Eercises 8 nd 8, two models nd re iven or revenue (in illions o dollrs per er) or lre corportion The model R ives projected nnul revenues rom to, with t correspondin to, nd R ives projected revenues i there is decrese in the rte o rowth o corporte sles over the period Approimte the totl reduction in revenue i corporte sles re ctull closer to the model R 6 D Proposl R Proposl t 6 Yer R 8 R 8 R 7 6t t 7 8t R 7 t R 7 t t 8 Modelin Dt The tle shows the totl receipts R nd totl ependitures E or the Old-Ae nd Survivors Insurnce Trust Fund (Socil Securit Trust Fund) in illions o dollrs The time t is iven in ers, with t correspondin to 99 (Source: Socil Securit Administrtion) t R E t R E () Use rphin utilit to it n eponentil model to the dt or receipts Plot the dt nd rph the model () Use rphin utilit to it n eponentil model to the dt or ependitures Plot the dt nd rph the model (c) I the models re ssumed to e true or the ers throuh 7, use intertion to pproimte the surplus revenue enerted durin those ers (d) Will the models ound in prts () nd () intersect? Eplin Bsed on our nswer nd news reports out the und, will these models e ccurte or lon-term nlsis? 8 Lorenz Curve Economists use Lorenz curves to illustrte the distriution o income in countr A Lorenz curve,, represents the ctul income distriution in the countr In this model, represents percents o milies in the countr nd represents percents o totl income The model represents countr in which ech mil hs the sme income The re etween these two models, where, indictes countr s income inequlit The tle lists percents o income or selected percents o milies in countr () Use rphin utilit to ind qudrtic model or the Lorenz curve () Plot the dt nd rph the model (c) Grph the model How does this model compre with the model in prt ()? (d) Use the intertion cpilities o rphin utilit to pproimte the income inequlit

10 SECTION 7 Are o Reion Between Two Curves 8 Proit The chie inncil oicer o compn reports tht proits or the pst iscl er were $89, The oicer predicts tht proits or the net ers will row t continuous nnul rte somewhere etween % nd % Estimte the cumultive dierence in totl proit over the ers sed on the predicted rne o rowth rtes 86 Are The shded reion in the iure consists o ll points whose distnces rom the center o the squre re less thn their distnces rom the edes o the squre Find the re o the reion Fiure or 86 Fiure or Mechnicl Desin The surce o mchine prt is the reion etween the rphs o nd 8 k (see iure) () Find k i the prol is tnent to the rph o () Find the re o the surce o the mchine prt 88 Buildin Desin Concrete sections or new uildin hve the dimensions (in meters) nd shpe shown in the iure (, ) = 6 () Find the re o the ce o the section superimposed on the rectnulr coordinte sstem () Find the volume o concrete in one o the sections multiplin the re in prt () meters (c) One cuic meter o concrete weihs pounds Find the weiht o the section 89 Buildin Desin To decrese the weiht nd to id in the hrdenin process, the concrete sections in Eercise 88 oten re not solid Rework Eercise 88 to llow or clindricl openins such s those shown in the iure m 8 m + Rottle Grph m 6 = (, ) True or Flse? In Eercises 9 9, determine whether the sttement is true or lse I it is lse, eplin wh or ive n emple tht shows it is lse 9 I the re o the reion ounded the rphs o nd is, then the re o the reion ounded the rphs o h C nd k C is lso 9 I d A, then d A 9 I the rphs o nd intersect midw etween nd, then d 9 Are Find the re etween the rph o sin nd the line sements joinin the points, nd 7 s 6,, shown in the iure (, ) 6 π Fiure or 9 Fiure or 9 9 Are Let > nd > Show tht the re o the ellipse 7 6 π (, is (see iure) ( π + Putnm Em Chllene 9 The horizontl line c intersects the curve in the irst qudrnt s shown in the iure Find c so tht the res o the two shded reions re equl = c = This prolem ws composed the Committee on the Putnm Prize Competition The Mthemticl Assocition o Americ All rihts reserved = m 6 (, ) = + 6 = (, ) Rottle Grph