' ' Š # # ' " # # # Section 5.3 The Definite Integral 275. œ x dx 1 dx x dx 1 dx. œ " # œ x dx 1 dx œ (3 1) œ ( 1 2) œ (see parts (a) and (b) above).
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1 Sectio. The Defiite Itegral 7 ( ) t dt t dt t dt t dt 6. av(f) Š at t dt Š ( ) ( ) Š Š. ( ) ( ) d ( ) d 6. (a) av(g) Š akk d d d d d ( ) Š ( ( )) Š ( ). () av(g) ˆ akk d ( ) d Š d d ( ). ( ) k k (c) av(g) Š akk d a d akk d ( ) (see parts (a) ad () aove). 6. (a) av(h) Š k k d ( ) d d. ( ) ( ) Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.
2 76 Chapter Itegratio () av(h) ˆ k kd d Š. ( ) (c) av(h) Š k k d Œ k k d k k d ˆ ˆ aove). (see parts (a) ad () k Ö a aa aa kaa k a a c a a k c a a k k k a Ä_ m mä 6. Cosider the partitio P that sudivides the iterval Òa, Ó ito suitervals of width a ad let c e the right edpoit of each suiterval. So the partitio is P a, a, a,..., a ad c a. We get the Riema sum f c c c a. As ad P this epressio remais ca a. Thus, c d ca a. a k k Ö k a ˆ k ˆ k 8 k 8 a a k k k k k a a 6. Cosider the partitio P that sudivides the iterval Ò, Ó ito suitervals of width ad let c e the right edpoit of each suiterval. So the partitio is P,,,..., ad c k. We get the Riema sum f c k. As Ä_ ad mp mä the epressio has the value 6. Thus, d Cosider the partitio P that sudivides the iterval Òa, Ó ito suitervals of width a ad let c e the right k a aa aa kaa Ö k a k ˆ Š k a a a k a a ak a a kaa Š k k k k a a k k a a a a a a a Œ a a a a a a a a k k k edpoit of each suiterval. So the partitio is P a, a, a,..., a ad c a. We get the Riema sum f c c a a aa a a aa aa Ä_ m mä a a a a a a aaa aa a aaa aa a As ad P this epressio has value a a a a a a a a a a a a. Thus, d. a 66. Cosider the partitio P that sudivides the iterval Ò, Ó ito suitervals of width ad let c e the right edpoit of each suiterval. So the partitio is P Ö,,,..., ad Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley. k
3 Sectio. The Defiite Itegral 77 k k k k k k k k k k k k Š ˆ a a a k k k k a a a. As Ä_ ad mp mä this epressio has value 6. Thus, a d. 6 c k. We get the Riema sum fac Š ˆ ˆ a k k k k k k k k k a a a Š k k k k 6a 7a a 8. As Ä_ ad mp mä this epressio has value Thus, a d Cosider the partitio P that sudivides the iterval Ò, Ó ito suitervals of width ad let c e the right edpoit of each suiterval. So the partitio is P Ö,,,..., ad c k. We get the Riema sum fac Š ˆ ˆ a k k a k ˆ ˆ k k k k 6k k 8k 6 8 Š k k Œ k k k k k k 6 a a a a a a a Š 68. Cosider the partitio P that sudivides the iterval Ò, Ó ito suitervals of width ad let c e the right edpoit of each suiterval. So the partitio is P Ö,,,..., ad c k. We get the Riema sum f c c k 6. As Ä _ ad mp m Ä this epressio has value 68. Thus, d. k a aa aa k a a a a k a a a k ˆ k Š k k k a a ka a ak a a k a a a a a a a a a a a a k a k k k k k k k a a a a a aa a a a a a a a Š 69. Cosider the partitio P that sudivides the iterval Òa, Ó ito suitervals of width a ad let c e the right edpoit of each suiterval. So the partitio is P Öa, a, a,..., a ad c a. We get the Riema sum f c c a k Š Œ a a a a a a a a a a a a aa a a a aaa a a a a a a aa a a a a a. As Ä_ ad mp mä this epressio has value a aa aa a. Thus, d. k k Ö k a k a k ˆ Š ˆ Œ k k k k k k k k 7. Cosider the partitio P that sudivides the iterval Ò, Ó ito suitervals of width ad let c e the right edpoit of each suiterval. So the partitio is P,,,..., ad c k. We get the Riema sum f c c c k k k k Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.
4 78 Chapter Itegratio a a a Š. As Ä_ ad mp mä this epressio has value. Thus, a d. 7. To fid where, let Ê ( ) Ê or. If, the Ê a ad maimize the itegral. 7. To fid where Ÿ, let Ê a Ê or È. By the sig graph,, we ca see that Ÿ o Èß È Ê a È ad È È È miimize the itegral. 7. f() is decreasig o [ß] Ê maimum value of f occurs at Ê ma f f() ; miimum value of f occurs at Ê mi f f(). Therefore, ( ) mi f Ÿ d Ÿ ( ) ma f Ê Ÿ d Ÿ. That is, a upper oud ad a lower oud. (.).. (.) Ÿ Ÿ Ê Ÿ Ÿ... 9 Ÿ Ÿ. Ê Ÿ Ÿ 7. See Eercise 7 aove. O [ß.], ma f, mi f.8. Therefore (. ) mi f Ÿ f() d Ÿ (. ) ma f Ê Ÿ d Ÿ. O [.ß], ma f.8 ad mi f.. Therefore (.) mi f d (.) ma f d. The d d d. 7. Ÿ si a Ÿ for all Ê ( )( ) Ÿ si a d Ÿ ( )() or si d Ÿ Ê si d caot equal. 76. f() È 8 is icreasig o [ß] Ê ma f f() È 8 ad mi f f() È 8 È. Therefore, ( ) mi f Ÿ È 8 d Ÿ ( ) ma f Ê È Ÿ È 8 d Ÿ. 77. If f() o [aß ], the mi f ad ma f o [aß]. Now, ( a) mi f Ÿ f() d Ÿ ( a) ma f. The a Ê a Ê ( a) mi f Ê f() d. 78. If f() Ÿ o [aß], the mi f Ÿ ad ma f Ÿ. Now, ( a) mi f Ÿ f() d Ÿ ( a) ma f. The a Ê a Ê ( a) ma f Ÿ Ê f() d Ÿ. a a 79. si Ÿ for Ê si Ÿ for Ê (si ) d Ÿ (see Eercise 78) Ê si d d Ÿ a si d d si d Š si d. Thus a upper oud is. Ê Ÿ Ê Ÿ Ê Ÿ a Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.
5 Sectio. The Defiite Itegral 79 ß ß Ê Ê 7 Š 6 8. sec o ˆ ß Ê sec Š o ˆ ß Ê sec Š d (see Eercise 77) sice [ ] is cotaied i ˆ sec d Š d sec d Š Ê sec d d d Ê sec d ( ) Ê sec d. Thus a lower oud 7 is 6. a a a a av(f) d ( a)k ( a) f() d f() d. a a a a 8. Yes, for the followig reasos: av(f) f() d is a costat K. Thus av(f) d K d K( a) Ê 8. All three rules hold. The reasos: O ay iterval [aß ] o which f ad g are itegrale, we have: a a a a a a a a a (a) av(f g) [f() g()] d f() d g() d f() d g() d av(f) av(g) a a a a a a () av(kf) kf() d k f() d k f() d k av(f) a a a a a a (c) av(f) f() d Ÿ g() d sice f() Ÿ g() o [aß], ad g() d av(g). Therefore, av(f) Ÿ av(g). 8. (a) U ma? ma? á ma? where ma f( ), ma f( ), á, ma f( ) sice f is icreasig o [aß]; L mi? mi? á mi? where mi f( ), mi f( ), á, mi f( c) sice f is icreasig o [aß]. Therefore U L (ma mi )? (ma mi )? á (ma mi )? (f() f( ))? (f( ) f( ))? á (f( ) f( c))? (f( ) f( ))? (f() f(a))?. () U ma? ma? á ma? where ma f( ), ma f( ), á, ma f( ) sice f is icreasig o[aß]; L mi? mi? á mi? where mi f( ), mi f( ), á, mi f( c) sice f is icreasig o [aß]. Therefore U L (ma mi )? (ma mi )? á (ma mi )? (f() f( ))? (f( ) f( ))? á (f( ) f( c))? Ÿ (f() f( ))? ma (f( ) f( ))? ma á (f( ) f( c))? ma. The U L Ÿ (f( ) f( ))? ma (f() f(a))? ma kf() f(a) k? ma sice f() f(a). Thus lim (U L) lim (f() f(a))? ma, sice? ma lp l. lpl Ä lpl Ä 8. (a) U ma? ma? á ma? where ma f( ), ma f( ), á, ma f( c ) sice f is decreasig o [aß ]; L mi? mi? á mi? where mi f( ), mi f( ) ß á, mi f( ) sice f is decreasig o [aß ]. Therefore U L (ma mi )? (ma mi )? á (ma mi )? (f() f( ))? (f( ) f( ))? á (f( c ) f( ))? (f( ) f( ))? (f(a) f())?. Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.
6 8 Chapter Itegratio () U ma? ma? á ma? where ma f( ), ma f( ), á, ma f( c ) sice f is decreasig o[aß]; L mi? mi? á mi? where mi f( ), mi f( ), á, mi f( ) sice f is decreasig o [aß]. Therefore U L (ma mi )? (ma mi )? á (ma mi )? (f() f( ))? (f( ) f( ))? á (f( c ) f( ))? Ÿ (f() f( ))? ma (f(a) f()? ma kf() f(a) k? ma sice f() Ÿ f(a). Thus lim (U L) lim kf() f(a) k? ma, sice? ma lp l. lpl Ä lpl Ä 8. (a) Partitio ß ito suitervals, each of legth? with poits,?,?, á,?. Sice si is icreasig o ß, the upper sum U is the sum of the areas of the circumscried rectagles of areas f( )? (si? )?, f( )? (si? )?, á, f( )? cos? cosˆ ˆ? si (si? )?. The U (si? si? á si? )??? cos cos ˆ ˆ ˆ cos cos ˆ cos cos ˆ si si si Š ˆ () The area is si d lim. Ä_ Î cos cos ˆ cos si Š 86. (a) The area of the shaded regio is m which is equal to L. i () The area of the shaded regio is M which is equal to U. i i i i i (c) The area of the shaded regio is the differece i the areas of the shaded regios show i the secod part of the figure ad the first part of the figure. Thus this area is U L. 87. By Eercise 86, U L M m where M maö fa o the ith suiterval ad i i i i i i i i a a i i i i i i i i i a a i i m miö f o the ith suiterval. Thus U L M m provided for each i ßÞÞÞ,. Sice a the result, U L a follows. 88. The car drove the first miles i hours ad the secod miles i hours, which meas it drove miles i 8 hours, for a average of mi/hr 7. mi/hr. I terms of average values of fuctios, the fuctio whose average value we seek is 8, Ÿ t Ÿ v(t), ad the average value is, Ÿ 8 ()() ()() Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.
7 89-9. Eample CAS commads: Maple: with( plots ); with( Studet[Calculus] ); f := -> -; a := ; := ; N :=[,,, ]; P := [seq( RiemaSum( f(), =a.., partitio=, method=radom, output=plot ), =N )]: display( P, isequece=true ); Sectio. The Defiite Itegral Eample CAS commads: Maple: with( Studet[Calculus] ); f := -> si(); a := ; := Pi; plot( f(), =a.., title=9(a) (Sectio.) ); N := [,, ]; () for i N do Xlist := [ a+.*(-a)/*i i=.. ]; Ylist := map( f, Xlist ); ed do: for i N do (c) Avg[] := evalf(add(y,y=ylist)/ops(ylist)); ed do; avg := FuctioAverage( f(), =a.., output=value ); evalf( avg ); FuctioAverage(f(),=a..,output=plot); (d) fsolve( f()=avg, =. ); fsolve( f()=avg, =. ); fsolve( f()=avg[], =. ); fsolve( f()=avg[], =. ); Eample CAS commads: Mathematica: (assiged fuctio ad values for a,, ad may vary) Sums of rectagles evaluated at left-had edpoits ca e represeted ad evaluated y this set of commads Clear[, f, a,, ] {a, }={, }; =; d = ( a)/; f = Si[] ; vals =Tale[N[], {, a, d, d}]; yvals = f /. Ä vals; oes = MapThread[Lie[{{,},{, },{, },{, }]&,{vals, vals d, yvals}]; Plot[f, {, a, }, Epilog Ä oes]; Sum[yvals[[i]] d, {i,, Legth[yvals]}]//N Sums of rectagles evaluated at right-had edpoits ca e represeted ad evaluated y this set of commads. Clear[, f, a,, ] {a, }={, }; =; d = ( a)/; f = Si[] ; Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.
8 8 Chapter Itegratio vals =Tale[N[], {, a d,, d}]; yvals = f /. Ä vals; oes = MapThread[Lie[{{,},{, },{, },{, }]&,{vals d,vals, yvals}]; Plot[f, {, a, }, Epilog Ä oes]; Sum[yvals[[i]] d, {i,,legth[yvals]}]//n Sums of rectagles evaluated at midpoits ca e represeted ad evaluated y this set of commads. Clear[, f, a,, ] {a, }={, }; =; d = ( a)/; f = Si[] ; vals =Tale[N[], {, a d/, d/, d}]; yvals = f /. Ä vals; oes = MapThread[Lie[{{,},{, },{, },{, }]&,{vals d/, vals d/, yvals}]; Plot[f, {, a, },Epilog Ä oes]; Sum[yvals[[i]] d, {i,, Legth[yvals]}]//N. THE FUNDAMENTAL THEOREM OF CALCULUS. ( ) d c d a () a( ) ( ) 6 c. ˆ ( ) d Š () Š ( ) c. a d a d Š Š a a a a. a d Š a a Š ( ) ( ) c a ( ). Š d Š Š 8 () () () a d Š () Š ( ) ( ) c ( ) 7. ˆ È d ˆ Î 8. d ˆ ( ) Î& Î& Î 9. sec d [ ta ] ˆ ta ˆ ( ta ) È È Î. a cos d [ si ] asi a si Î Î. csc ) cot ) d ) [ csc )] ˆ csc ˆ ˆ csc ˆ È Š È Î Î. sec u ta u du [ sec u] sec ˆ sec () () Î Î Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.
9 Sectio. The Fudametal Theorem of Calculus 8 Î Î Î cos t. dt ˆ cos t dt t si t ˆ () si () ˆ ˆ si ˆ Î Î cos t. dt ˆ cos t dt t si t cî cî Î Î È 6 6 ˆ ˆ si ˆ ˆ ˆ si ˆ si si ˆ Î Î Î. ta d asec d [ta ] ˆ ta ˆ ata a Î6 Î6 Î6 6. asec ta d asec sec ta ta d asec sec ta d Î6 ˆ ˆ ˆ ˆ a È [ ta sec ] ta sec ta sec 7. si d cos ˆ cos ˆ ˆ cos a Î8 Î8 È 8 Î Î cî t cî Î t Î 8. ˆ sec t dt a sec t t dt ta t Š ta ˆ Š ta ˆ (( ) ) Š Š È È ˆ ˆ r ( ) 8 9. (r ) dr ar r dr r r Š ( ) ( ) Š È È. (t ) t dt t t È a È at t t dt t t Š È È Š È Š È È È È Š È Š È Š È Š È ) Š È u u & u u 6 È 6 6 Š È ( ( ) ). È Š & du È Š u du u Š () c c c y y y ac ac c y c c ac ac. dy ay y dy y Š Š È È È. s Ès ds s ˆ Î s Î ds s È È Š È È ÈÈ 8 ˆ ˆ s ÉÈ Î Î Î Î Î Î Î Î Î. d d ˆ d Î Î Î Î Î Î Î Î Î Š 8 a a8 8 a a8 Š a a a a 7 si sicos. d d cos d si asi a ˆ si ˆ Î si Î si Î Î Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.
10 8 Chapter Itegratio Î Î Î 6. acos sec d acos sec d ˆ cos sec d Î Î ˆ cos sec d si ta ˆ si ˆ ˆ taˆ ˆ si a a taa 9È k k d k k d k k d d d Š Š 6 c c c Î Î ( ) 8. acos kcos k d (cos cos ) d (cos cos ) d cos d [si ] Î si si È È È 9. (a) cos t dt [si t] si È si si È d d Î Ê Œ cos t dt ˆ si È cos È ˆ cos È È È d d () Œ cos t dt ˆ cos È ˆ ˆ È ˆ cos È ˆ Î d d d cos È È si si si d d d d. (a) t dt ct d si Ê Œ t dt asi si cos d d () Œ t dt a si ˆ (si ) si cos d si d Î t t t Î Î & dt dt t. (a) Èu du u du u d t t È d a Ê Œ u du ˆ t t t d d () Œ Èu du Èt ˆ at t at t dt & dt ta ) ta ) d d d) d) ta ). (a) sec y dy [ta y] ta (ta )) ta (ta )) Ê sec y dy (ta (ta ))) Œ asec (ta )) sec ) d d () Œ sec y dy asec (ta ) ˆ (ta ) ) ) asec (ta )) sec ) d) d) ta ). y È t dy dt Ê È dy. y dt Ê, d t d È. y si t dt dy d Î È si t dt Ê si (si ) d Š ˆ È ˆ ˆ È ˆ d dy d d d d d 6. y si t dt Ê Œ si t dt si t dt si a a si t dt 6 si si t dt t t dy ct t d 7. y dt dt Ê d si È Î Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.
11 Sectio. The Fudametal Theorem of Calculus 8 dy d d d 8. y Œ at dt Ê Œ at dt Œ at dt a Œ at dt si Èt d Èsi d Ècos kcos k cos dt dy d cos cos 9. y, k k Ê ˆ (si ) (cos ) sice kk dt dy. y Ê ˆ ˆ d (ta ) ˆ asec ta t d ta d sec. Ê ( ) Ê or ; Area a d a d a d ( ) ( ) Š Š ( ) Š ( ) ( ) Š Š Š ( ) 8 Š Š Š. Ê Ê ; ecause of symmetry aout the y-ais, Area Œ a d a d Š c d c d c aa () a () aa () a () d (6). Ê a Ê ( )( ) Ê,, or ; Area a d a d Š Š Š Š Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.
12 86 Chapter Itegratio Î Î Î Î. Ê ˆ Ê or Î Î Ê or Ê or Ê or ; ) Area ˆ Î d ˆ Î Î d ˆ d c ) Î Î Î Î Î ( ) Š () Š ( ) Î Î Š () Š () Î 8 Î 8 Š (8) Š () ˆ. The area of the rectagle ouded y the lies y, y,, ad is. The area uder the curve y cos o [ ß] is ( cos ) d [ si ] ( si ) ( si ). Therefore the area of the shaded regio is The area uder the curve y si o Î6 & Î 6 6 6ß 6 is si d [ cos ] Î6 Î ˆ ˆ È È Š È È The area of the rectagle ouded y the lies,, y si si, ad y is ˆ cos cos. Therefore the area of the shaded regio is. 7. O ß : The area of the rectagle ouded y the lies y È, y, ), ad ) is È ˆ È. The area etwee the curve y sec ) ta ) ad y is sec ) ta ) d ) [ sec )] cî Î ˆ È È Î ) ) Î ) ) ) ) È È ( sec ) sec ˆ È. Therefore the area of the shaded regio o ß is Š È. O ß : The area of the rectagle ouded y,, y È, ad y is È ˆ ) ). The area uder the curve y sec ta is sec ta d [sec ] sec sec. Therefore the area of the shaded regio o ß is Š È. Thus, the area of the total shaded regio is È È È Š È Š È. 8. The area of the rectagle ouded y the lies y, y, t, ad t is ˆ ˆ. The area uder the curve y sec t o ß is sec t dt [ta t] ta ta ˆ Î. The area cî uder the curve y t o [ß ] is t a t dt t Š Š. Thus, the total area uder the curves o ß is. Therefore the area of the shaded regio is ˆ. dy t d t 9. y dt Ê ad y( ) dt Ê (d) is a solutio to this prolem. dy c c d c. y sec t dt Ê sec ad y( ) sec t dt Ê (c) is a solutio to this prolem. dy d. y sec t dt Ê sec ad y() sec t dt Ê () is a solutio to this prolem. Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.
13 dy Sectio. The Fudametal Theorem of Calculus 87. y dt Ê ad y() dt Ê (a) is a solutio to this prolem. t d t È. y sec t dt. y t dt Î Î. Area ˆ h ˆ h h d h cî h ˆ h ˆ Œhˆ Œhˆ ˆ ˆ h h h h h h 6 6 cî h Îk Îk k k 6. k Ê oe arch of y si k will occur over the iterval ß Ê the area si k d cos k cos ˆ k ˆ ˆ cos () k k k k dc 7. Î Ê c Î t dt Î t È; c() c() È È 9. d È 8. r Š d Š d ˆ Š Š ( ) ( ) ( ) ( ) ˆ. or 9. (a) t Ê T 8 È 7 F; t 6 Ê T 8 È 6 76 F; È t Ê T 8 8 F Î () average temperatuve Š 8 È t dt 8t a t Î Î Š 8aa Š 8aa 7 F 6. (a) t Ê H È a Î ft; t Ê H È a Î È È.7 ft; t 8 Ê H È8a8 Î ft 8 Î Î Î () average height t t dt t t Š È a 8 8 Î Î Î Î Š a8 a8 Š a a 9.67 ft d d 6. f(t) dt Ê f() d f(t) dt d a d d 6. f(t) dt cos Ê f() f(t) dt cos si Ê f() cos () () si () 9 w 9 9 w 9 t () t 6. f() dt Ê f () Ê f () ; f() dt ; L() ( ) f() ( ) 8 Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.
14 88 Chapter Itegratio 6. g() sec (t ) dt Ê g () asec a () sec a Ê g ( ) ( ) sec a( ) w w a ; g( ) sec (t ) dt sec (t ) dt ; L() ( ( )) g( ) ( ) 6. (a) True: sice f is cotiuous, g is differetiale y Part of the Fudametal Theorem of Calculus. () True: g is cotiuous ecause it is differetiale. w (c) True, sice g () f(). ww w (d) False, sice g () f (). w ww w (e) True, sice g () ad g () f (). ww w ww (f) False: g () f (), so g ever chages sig. w w w (g) True, sice g () f() ad g () f() is a icreasig fuctio of (ecause f () ). 66. Let a â e ay partitio of Òa, Óad let F e ay atiderivative of f. (a) Fa Fa i i i F a F a F a F a F a F a â F a F a F a F a F a F a F a F a F a âf a F a F a F a F a F a Fa a () Sice F is ay atiderivative of f o Òa, Ó Ê F is differetiale o Òa, Ó Ê F is cotiuous o Òa, Ó. Cosider ay suiterval Ò, Ói Òa, Ó, the y the Mea Value Theorem there is at least oe umer c i Ð, Ñ such that i i i i i w i i i i i i i i i? i i i i F a F a Faca fc a a fc a. Thus F a Fa a F a F a fc a?. i i i (c) Takig the limit of FaFaa faci? i we otai lim afafaa lim Œ faci? i a Ê F a Fa a f a d i mpmä mpmä i Eample CAS commads: Maple: with( plots ); f := -> ^-*^+*; a := ; := ; F := uapply( it(f(t),t=a..), ); (a) p := plot( [f(),f()], =a.., leged=[y = f(),y = F()], title=67(a) (Sectio.) ): p; df := D(F); () q := solve( df()=, ); pts := [ seq( [,f()], =remove(has,evalf([q]),i) ) ]; p := plot( pts, style=poit, color=lue, symolsize=8, symol=diamod, leged=(,f()) where F ()= ): display( [p,p], title=8() (Sectio.) ); icr := solve( df()>, ); (c) decr := solve( df()<, ); df := D(f); (d) p := plot( [df(),f()], =a.., leged=[y = f (),y = F()], title=67(d) (Sectio.) ): p; q := solve( df()=, ); Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.
15 Sectio. The Fudametal Theorem of Calculus 89 pts := [ seq( [,F()], =remove(has,evalf([q]),i) ) ]; p := plot( pts, style=poit, color=lue, symolsize=8, symol=diamod, leged=(,f()) where f ()= ): display( [p,p], title=8(d) (Sectio.) ); 7-7. Eample CAS commads: Maple: a := ; u := -> ^; f := -> sqrt(-^); F := uapply( it( f(t), t=a..u() ), ); df := D(F); () cp := solve( df()=, ); solve( df()>, ); solve( df()<, ); df := D(dF); (c) solve( df()=, ); plot( F(), =-.., title=7(d) (Sectio.) ); 7. Eample CAS commads: Maple: f := `f`; q := Diff( It( f(t), t=a..u() ), ); d := value( q ); 76. Eample CAS commads: Maple: f := `f`; q := Diff( It( f(t), t=a..u() ),, ); value( q ); Eample CAS commads: Mathematica: (assiged fuctio ad values for a, ad may vary) For trascedetal fuctios the FidRoot is eeded istead of the Solve commad. The Map commad eecutes FidRoot over a set of iitial guesses Iitial guesses will vary as the fuctios vary. Clear[, f, F] {a, }= {, }; f[_] = Si[] Cos[/] F[_] = Itegrate[f[t], {t, a, }] Plot[{f[], F[]},{, a, }] /.Map[FidRoot[F[]==, {, }] &,{,,, 6}] /.Map[FidRoot[f[]==, {, }] &,{,,,, 6}] Slightly alter aove commads for 7-8. Clear[, f, F, u] a=; f[_] = u[_] = F[_] = Itegrate[f[t], {t, a, u()}] /.Map[FidRoot[F[]==,{, }] &,{,,, }] /.Map[FidRoot[F[]==,{,}] &,{,,, }] Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.
16 9 Chapter Itegratio After determiig a appropriate value for, the followig ca e etered = ; Plot[{F[], {, a, }]. INDEFINTE INTEGRALS AND THE SUBSTITUTION RULE Let u Ê du d Ê du d a a d u du u du u C C. Let u 7 Ê du 7 d Ê 7 du d 7È7 d Î 7 7 d Î a 7u du Î Î Î u du u C a7 C. Let u Ê du d Ê du d 7 a a d u du u du u C C a a. Let u Ê du d Ê du d d d u du u du u C C. Let u Ê du a6 d a d Ê du a d a a a d u du u du u C C 6. Let u Ê du d Ê du d È È È Î Î È È ˆ È Î Î Î Î d ˆ È d u du u du u C ˆ È C 7. Let u Ê du d Ê du d si d si u du cos u C cos C 8. Let u Ê du d Ê du d si a d si u du cos u C cos C 9. Let u t Ê du dt Ê du dt sec t ta t dt sec u ta u du sec u C sec t C t t t t t t. Let u cos Ê du si dt Ê du si dt ˆ cos ˆ si dt u du u C ˆ cos C. Let u r Ê du r dr Ê du 9r dr 9r dr Î Î Î È r u du ()u C 6 a r C. Let u y y Ê du ay 8y dy Ê du ay y dy y y y y dy a a u du u C ay y C Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.
17 Î Î. Let u Ê du d Ê du È d Sectio. Idefiite Itegrals ad the Sustitutio Rule 9 È si ˆ Î d si u du ˆ u si u C ˆ Î si ˆ Î C. Let u Ê du d 6 si ˆ C cos ˆ u d cos au du cos au du ˆ si u C si ˆ C. (a) Let u cot ) Ê du csc ) d ) Ê du csc ) d) u u csc ) cot ) d) u du Š C C cot ) C () Let u csc ) Ê du csc ) cot ) d ) Ê du csc ) cot ) d) u u csc ) cot ) d) u du Š C C csc ) C 6. (a) Let u 8 Ê du d Ê du d Î Î Î Š du u du ˆ u C u C È 8 C d È 8 Èu È Î d Ê Ê È8 () Let u 8 du ( 8) () d du du uc È8C d È 8 7. Let u s Ê du ds Ê du ds È s ds Èu ˆ du Î u du ˆ ˆ Î u Î C ( s) C 8. Let u s Ê du ds Ê du ds Î Î ds ˆ du u du ˆ ˆ u C Ès C Ès Èu 9. Let u ) Ê du ) d ) Ê du ) d) ) È ) d) È u ˆ du Î u du ˆ ˆ &Î u C a ) &Î C. Let u 7 y Ê du 6y dy Ê du y dy yè7 y dy Èu ˆ du u du ˆ ˆ u C a7 y C È È È du Ȉ È u u È. Let u Ê du d Ê du d Î Î Î d C C. Let u z Ê du dz Ê du dz cos (z ) dz (cos u) ˆ du cos u du si u C si (z ) C. Let u Ê du d Ê du d a ˆ sec ( ) d sec u du sec u du ta u C ta ( ) C. Let u ta Ê du sec d ta sec d u du u C ta C Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.
18 9 Chapter Itegratio. Let u si ˆ Ê du cos ˆ d Ê du cos ˆ d & si ˆ cos ˆ & d u ( du) ˆ u C si ˆ C 6 6. Let u ta ˆ Ê du sec ˆ d Ê du sec ˆ d ( ta ˆ sec ˆ ( ) ) d u ( du) ˆ u C ta ˆ C 8 r r 8 6 r & & & u r Let u Ê du dr Ê 6 du r dr r Š dr u (6 du) 6 u du 6 Š C Š C & r Š & & r u r Š Š 8. Let u 7 Ê du r dr Ê du r dr r 7 dr u ( du) u du C 7 C Î Î Î Î Î Î 9. Let u Ê du d Ê du d si ˆ d (si u) ˆ du si u du ( cos u) C cos ˆ C. Let u csc ˆ v Ê du csc ˆ v cot ˆ v dv Ê du csc ˆ v cot ˆ v dv csc ˆ v cot ˆ v dv du u C csc ˆ v C. Let u cos (t ) Ê du si (t ) dt Ê du si (t ) dt si (t ) du cos (t ) u u cos (t ) dt C C. Let u sec z Ê du sec z ta z dz sec z ta z Èsec z Î Î È Èu dz du u du u C sec z C t t t t t. Let u t Ê du t dt Ê du dt cos ˆ dt (cos u)( du) cos u du si u C si ˆ C È Î Î Èt. Let u t t Ê du t dt Ê du dt Èt cos ˆ Èt dt (cos u)( du) cos u du si u C si ˆ Èt C. Let u si Ê du ˆ cos ˆ d ) Ê du cos d) ) ) ) ) ) ) ) ) ) ) si cos d u du u C si C 6. Let u csc È Ê du csc È cot È ) Š ) ) Š d ) Ê du cot È) csc È) d) cos È È ) È ) ) d ) cot È) csc È) d) du u C csc È) C C È) si È) È) si È) a ˆ ˆ 6 a 7. Let u t Ê du t dt Ê du t dt t t dt u du u C t C Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.
19 8. Let u Ê du d Sectio. Idefiite Itegrals ad the Sustitutio Rule 9 É d É d É d È u du u du u C ˆ C Î Î Î & 9. Let u Ê du d Î Î Î É d Èu du u du u C ˆ C. Let u Ê du d Ê du d É d É d È u du u du u C ˆ C Î Î Î 9 9. Let u Ê du d Ê du d É d É d É d È u du u du u C ˆ C Î Î Î Î Î Î È Èu. Let u Ê du d Ê du d É d d du u du u C a C. Let u. The du d ad u. Thus a d au u du au u du u u C a a C. Let u. The du d ad adu d ad u. Thus È d a uèu adu a a uˆ Î u Î Î Î 8 Î Î 8 Î du u u du u u C a a C. Let u. The du d ad adu d ad u. Thus a a d a 7a a C au u adu au u u du u u u C Î Î 6. Let u. The du d ad u. Thus a a d au u du ˆ Î Î u u du 7 Î Î 7 Î Î 7 7 u u C a a C 7. Let u. The du d ad du d ad u. Thus È d au Èu du a a a Î Î &Î Î &Î Î &Î & & & u u du u u C u u C C & 8. Let u Êdu d ad u. So B È Î Î d au Èu du u u du &Î Î &Î & & u u C a a C a Î a u C a C 9. Let u Ê du d ad du d. Thus d d u du u du a a. Let u Ê du d ad u. Thus d a d u au du u u du a u u C a a C Î Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.
20 9 Chapter Itegratio. (a) Let u ta Ê du sec d; v u Ê dv u du Ê 6 dv 8u du; w v Ê dw dv 8 ta sec 8u 6 dv 6 dw 6 ata au ( v) w v C u ta C d du 6 w dw 6w C C () Let u ta Ê du ta sec d Ê 6 du 8 ta sec d; v u Ê dv du 8 ta sec 6 du 6 dv ta a ( u) v v u ta d C C C (c) Let u ta Ê du ta sec d Ê 6 du 8 ta sec d 8 ta sec 6 du 6 6 ta a u u ta d C C. (a) Let u Ê du d; v si u Ê dv cos u du; w v Ê dw v dv Ê dw v dv È si ( ) si ( ) cos ( ) d È si u si u cos u du vè v dv Î Î Î È Î w dw w C av C asi u C asi () C () Let u si ( ) Ê du cos ( ) d; v u Ê dv u du Ê dv u du È si ( ) si ( ) cos ( ) d u È u du È Î v dv v dv ˆ Î Î Î Î a a ˆ v C v C u C si () C (c) Let u si ( ) Ê du si ( ) cos ( ) d Ê du si ( ) cos ( ) d È si ( ) si ( ) cos ( ) d È Î u du u du ˆ Î u C Î a si ( ) C. Let u (r ) 6 Ê du 6(r )() dr Ê du (r ) dr; v È u Ê dv du Ê dv du (r ) cos È(r ) 6 È(r ) 6 Èu 6 Èu cos Èu Èu dr Š ˆ du (cos v) ˆ dv si v C si Èu C si È(r ) 6 C 6. Let u cos È Ê du si È si ) ) Š ) Š d ) Ê du d) È È ) È ) si È) È) uî É cos È È Écos È È ) ) ) ) u C Écos È ) si du Î d ) d ) u du ˆ Î u C C. Let u t Ê du 6t dt Ê du t dt a s t at dt u ( du) ˆ u C u C at C; s whe t Ê ( ) C Ê 8 C Ê C Ê s t 6. Let u 8 Ê du d Ê du d Î Î y a 8 d u ( du) ˆ Î u Î C u C a 8 C; y whe Ê (8) C Ê C Ê y a 8 7. Let u t Ê du dt Î Î s 8 si ˆ t dt 8 si u du 8 ˆ u si u C ˆ t si ˆ t C; 6 ˆ ˆ s 8 whe t Ê 8 si C Ê C 8 9 Ê s ˆ t si ˆ t 9 t si ˆ t 9 Î Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.
21 Sectio. Idefiite Itegrals ad the Sustitutio Rule 9 8. Let u ) Ê du d) r cos ˆ d cos u du ˆ u si u C ˆ si ˆ ) ) ) ) C; 8 ) ˆ 8 8 ) ˆ ) r si r cos ) ˆ ) 8 ) ) 8 r whe Ê si C Ê C Ê r si Ê Ê 9. Let u t Ê du dt Ê du dt ds dt si ˆ t dt (si u)( du) cos u C cos ˆ t C ; ds at t ad we have cos ˆ ds C Ê C Ê cos ˆ t dt dt Ê s ˆ cos ˆ t dt (cos u ) du si u u C si ˆ t ˆ t C ; ˆ ˆ at t ad s we have si C Ê C Ê s si ˆ t t ( ) Ê s si ˆ t t 6. Let u ta Ê du sec d Ê du sec d; v Ê dv d Ê dv d dy d sec ta d u( du) u C ta C ; dy d at ad we have C Ê C Ê ta asec sec dy d Ê y asec d asec v ˆ dv ta v v C ta C ; at ad y we have () C Ê C Ê y ta 6. Let u t Ê du dt Ê du 6 dt s 6 si t dt (si u)( du) cos u C cos t C; at t ad s we have cos C Ê C Ê s cos t Ê s ˆ cos ( ) 6 m 6. Let u t Ê du dt Ê du dt v cos t dt (cos u)( du) si u C si ( t) C ; at t ad v 8 we have 8 () C Ê C 8 Ê v si ( t) 8 Ê s ( si ( t) 8) dt si u du 8t C cos ( t) 8t C ; at t ad s we have C Ê C Ê s 8t cos ( t) Ê s() 8 cos m 6. All three itegratios are correct. I each case, the derivative of the fuctio o the right is the itegrad o the left, ad each formula has a aritrary costat for geeratig the remaiig atiderivatives. Moreover, cos si C cos C Ê C C ; also cos C C Ê C C C. Î6 Î 6 ma ma 6. (a) Š V si t dt 6 V ˆ cos ( t) V [cos cos ] V ma [ ] () V È V È () 9 volts ma rms Î6 Î6 Î6 a ma a ma ˆ cos t avma a t si t a V V si ( ) si () a V ma Î ma ma ˆ ˆ ˆ ˆ ˆ 6 (c) V si t dt V dt ( cos t) dt ds dt Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.
22 96 Chapter Itegratio.6 SUBSTITUTION AND AREA BETWEEN CURVES. (a) Let u y Ê du dy; y Ê u, y Ê u È Î y dy u du Î u ˆ Î () ˆ Î () ˆ (8) ˆ () () Use the same sustitutio for u as i part (a); y Ê u, y Ê u È Î y dy u du Î u ˆ Î () c r È r dr Èu du Î u ˆ Î (). (a) Let u r Ê du r dr Ê du r dr; r Ê u, r Ê u () Use the same sustitutio for u as i part (a); r Ê u, r Ê u r È r dr Èu du c Î ta sec d u u du ta sec d u u du cî c. (a) Let u ta Ê du sec d; Ê u, Ê u () Use the same sustitutio as i part (a); Ê u, Ê u. (a) Let u cos Ê du si d Ê du si d; Ê u, Ê u cos si d u du cu d ( ) () a () Use the same sustitutio as i part (a); Ê u, Ê u cos si d u du t t dt u a u du (a) u t Ê du t dt Ê du t dt; t Ê u, t Ê u () Use the same sustitutio as i part (a); t Ê u, t Ê u t a t dt u du c 6. (a) Let u t Ê du t dt Ê du t dt; t Ê u, t È7 Ê u 8 È 7 8 Î ) Î t at dt u du ˆ ˆ Î u ˆ Î (8) ˆ Î () () Use the same sustitutio as i part (a); t È7 Ê u 8, t Ê u 8 Î cè 7 8 Î Î t at dt u du u du r dr u du a r 7. (a) Let u r Ê du r dr Ê du r dr; r Ê u, r Ê u () Use the same sustitutio as i part (a); r Ê u, r Ê u r & dr u du u () () a r ˆ ˆ 8 8 Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.
23 Sectio.6 Sustitutio ad Area Betwee Curves 97 Î Î 8. (a) Let u v Ê du v dv Ê du Èv dv; v Ê u, v Ê u Èv dv ˆ du u du a v u Î u Î () Use the same sustitutio as i part (a); v Ê u, v Ê u 9 9 Èv * 7 7 dv ˆ du v u a u ˆ Î 9 ˆ 8 7 È 9. (a) Let u Ê du d Ê du d; Ê u, Ê u È Î Î Î Î d du u du u () () È Èu () Use the same sustitutio as i part (a); È Ê u, È Ê u È d du cè È È u d Î Î Î Î È u du ()u () (9) È 9 9 *. (a) Let u 9 Ê du d Ê du d; Ê u 9, Ê u () Use the same sustitutio as i part (a); Ê u, Ê u 9 9 Î Î È d u du u du c È Î6 ( cos t) si t dt u du () () Š u Ê Ê Î ( cos t) si t dt u du () () Î6 Š u 6 6. (a) Let u cos t Ê du si t dt Ê du si t dt; t Ê u, t Ê u cos () Use the same sustitutio as i part (a); t u, t u cos t t t. (a) Let u ta Ê du sec dt Ê du sec dt; t Ê u ta ˆ, t Ê u ˆ t ta t sec dt u ( du) cu d Î ˆ t ta t sec dt u du cu d 8 c Î () Use the same sustitutio as i part (a); t Ê u, t Ê u c Î. (a) Let u si z Ê du cos z dz Ê du cos z dz; z Ê u, z Ê u cos z dz du È ˆ si z Èu () Use the same sustitutio as i part (a); z Ê u si ( ), z Ê u c cos z È si z Èu dz ˆ du si w dw & u du u cî ( cos w) ˆ c d ˆ Ê Ê Î si w dw u du u du ( cos w) ˆ. (a) Let u cos w Ê du si w dw Ê du si w dw; w Ê u, w Ê u () Use the same sustitutio as i part (a); w u, w u Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.
24 98 Chapter Itegratio &. Let u t t Ê du at dt; t Ê u, t Ê u Èt& t at dt Î u du Î u Î Î () () È dy Èy 6. Let u Èy Ê du ; y Ê u, y Ê u dy du u du u Èyˆ Èy u c d ˆ ˆ 6 7. Let u cos ) Ê du si ) d ) Ê du si ) d ); ) Ê u, ) Ê u cos ˆ 6 6 Î6 Î Î Î cos ) si ) d) u ˆ du u du Š c ˆ () 8. Let u ta ˆ ) Ê du sec ˆ ) d Ê 6 du sec ˆ ) d ; Ê u ta ˆ ) ) ), ) Ê u ta È Î c & cot ˆ ) sec ˆ ) & u d u (6 du) 6 Š 6 6 ) È u Î È () Î Î È Š È 9. Let u cos t Ê du si t dt Ê du si t dt; t Ê u cos, t Ê u cos * Î Î ( cos t) si t dt u ˆ du Î u du ˆ &Î &Î &Î u 9. Let u si t Ê du cos t dt Ê du cos t dt; t Ê u, t Ê u Î Î Î ( si t) cos t dt u du ˆ &Î u ˆ &Î &Î () ˆ (). Let u y y y Ê du a y y dy; y Ê u, y Ê u () () () 8 8 Î ) Î ay y y ay y dy u du Î u Î Î (8) () Î y 6y y 9 y y dy Î a a u du ˆ Î u Î Î () (9) ( ) 9 *. Let u y 6y y 9 Ê du ay y dy Ê du ay y dy; y Ê u 9, y Ê u Î Î. Let u ) Ê du ) d ) Ê du È) d ); ) Ê u, ) È Ê u È Î È cos ˆ d cos u ˆ du ˆ u si u ) ) ) ˆ si () t cî c u t si. Let u Ê du t dt; t Ê u, t Ê u t si ˆ dt si u du ˆ si u ˆ si ( ) ˆ si c È È Î Î Î Î c Î Î Î 6 u () (). Let u Ê du d Ê du d; Ê u, Ê u, Ê u A d d u du u du u du u du 6. Let u cos Ê du si d; Ê u, Ê u ( cos ) si d u du u Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.
25 Sectio.6 Sustitutio ad Area Betwee Curves Let u cos Ê du si d Ê du si d; Ê u cos ( ), Ê u cos Î Î Î Î Î &Î c A (si ) È cos d u ( du) u du u () () 8. Let u si Ê du cos d Ê du cos d; Ê u si ˆ, Ê u Because of symmetry aout, A (cos ) (si ( si )) d (si u) ˆ du si u du [ cos u] ( cos ) ( cos ) c Î cos 9. For the sketch give, a, ; f() g() cos si ; (cos ) A d ( cos ) d si [( ) ( )] Î Î Î Î Î ˆ (cos t). For the sketch give, a, ; f(t) g(t) sec t a si t sec t si t; Î Î Î Î È A sec t si t dt sec t dt si t dt sec t dt dt c Î c Î c Î c Î c Î sec t dt ( cos t) dt [ta t] [t ] È si t cî cî Î Î. For the sketch give, a, ; f() g() a ; & A a d ˆ ˆ c. For the sketch give, c, d ; f(y) g(y) y y ; A ay y dy y dy y dy y y ( ) ( ). For the sketch give, c, d ; f(y) g(y) ay y ay y y y y; A ay y y dy y dy y dy y dy y y y ˆ () (). For the sketch give, a, ; f() g() a ; & A a d ˆ ˆ c. We wat the area etwee the lie y, Ÿ Ÿ, ad the curve y, 78?= the area of a triagle (formed y y ad y ) with ase ad height. Thus, A Š d ()() ˆ We wat the area etwee the -ais ad the curve y, Ÿ Ÿ :6?= the area of a triagle (formed y, y, ad the -ais) with ase ad height. Thus, A d ()() 6 7. AREA A A A: For the sketch give, a ad we fid y solvig the equatios y ad y simultaeously for : Ê Ê ( )( ) Ê or so c c : f() g() a a Ê A a d Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.
26 Chapter Itegratio ˆ ( 8 9 ) 9 ; A: For the sketch give, a ad : f() g() a a Ê A a d ˆ ˆ 6 8 c 6 8 9; Therefore, AREA A A AREA A A A: For the sketch give, a ad : f() g() a a 8 Ê A a 8 d (8 6) 8; c 8 A: For the sketch give, a ad : f() g() a a 8 Ê A a8 d (6 8) 8; Therefore, AREA A A AREA A A A A: For the sketch give, a ad : f() g() ( ) a c Ê A a d ˆ ˆ 8 7 ; c 6 6 A: For the sketch give, a ad : f() g() a ( ) a Ê A a d ˆ 8 ˆ 9 8 ; c A: For the sketch give, a ad : f() g() ( ) a Ê A a d ˆ ˆ ; 9 ˆ Therefore, AREA A A A 9 9. AREA A A A A: For the sketch give, a ad : f() g() Š a c (8 ) ; Š a Ê A a d ( 8) ; A: For the sketch give, a ad we fid y solvig the equatios y ad y simultaeously for : Ê Ê ( )( ) Ê,, or so : f() g() Š a Ê A a d a A: For the sketch give, a ad : f() g() Ê A a d ˆ 9 ˆ 8 ˆ ; Therefore, AREA A A A. a, ; f() g() a Ê A a d ˆ 8 8 ˆ 8 8 c 8 ˆ Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.
27 Sectio.6 Sustitutio ad Area Betwee Curves. a, ; f() g() a ( ) c 7 Ê A a d ˆ 9 9 ˆ. a, ; f() g() 8 Ê A a8 d &. Limits of itegratio: Ê Ê ( ) Ê a ad ; f() g() a Ê A a d Limits of itegratio: Ê Ê ( ) Ê a ad ; f() g() a Ê A a d 6. Limits of itegratio: 7 Ê Ê ( )( ) Ê a ad ; f() g() a7 a Ê A a d c ˆ ˆ 6ˆ 7. Limits of itegratio: Ê Ê a a Ê ( )( )( )( ) Ê,,, ; f() g() a ad g() f() a c c c Ê A a d a d a d & & & ˆ ˆ 8 ˆ ˆ ˆ 8 ˆ 8 Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.
28 Chapter Itegratio 8. Limits of itegratio: È a Ê or È a Ê or a Ê a,, a; A Èa d Èa d ca Î a Î aa aa a Î Î a aa aa a È, Ÿ 9. Limits of itegratio: y Èk k ad È, 6 y 6 or y ; for Ÿ : È 6 Ê È 6 Ê ( ) 6 Ê 7 6 Ê ( )( 6) Ê, 6 (ut 6 is ot a solutio); for : È 6 Ê 6 Ê 6 Ê ( )( 9) Ê, 9; there are three itersectio poits ad 9 A ˆ 6 È d ˆ 6 È d ˆ È 6 d c * ( 6) Î ( 6) Î Î ( 6) ( ) 6 Î 6 Î Î 9 ˆ ˆ ˆ. Limits of itegratio:, Ÿ or y k k, Ÿ Ÿ for Ÿ ad : Ê 8 8 Ê 6 Ê ; for Ÿ Ÿ : Ê 8 8 Ê Ê ; y symmetry of the graph, A Š a d Š a d 8 ˆ ˆ 6. Limits of itegratio: c ad d ; f(y) g(y) y y Ê A y dy 9 8 y Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.
29 . Limits of itegratio: y y Ê (y )(y ) Ê c ad d ; f(y) g(y) (y ) y y y c Ê A ay y dy y ˆ ˆ Sectio.6 Sustitutio ad Area Betwee Curves. Limits of itegratio: y ad 6 y Ê y 6 y Ê y y Ê (y )(y ) Ê c ad d ; f(y) g(y) ˆ Š 6y y y y c y y & 6 6 ˆ Ê A ay y dy y ˆ ˆ 8. Limits of itegratio: y ad y Ê y y Ê y Ê (y )(y ) Ê c ad d ; f(y) g(y) a y y y ay Ê A ay dy y y ˆ ˆ c ˆ. Limits of itegratio: y y ad y y 6 Ê y y y y 6 Ê y y 6 Ê ay ay Ê c ad d ; f(y) g(y) ay yay y 6 y y 6 y c Ê A ay y 6 dy y 6y ˆ 9 8 ˆ Î 6. Limits of itegratio: y ad y Î Ê y y Ê c ad d ; Î f(y) g(y) a y y Ê A ˆ Î y y dy c & y &Î y y ˆ ˆ ˆ Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.
30 Chapter Itegratio 7. Limits of itegratio: y ad kykè y Ê y kykèy Ê y y y ay Ê y y y y Ê y y Ê ay ay Ê y or y È Ê y or y Ê y or y. È Sustitutio shows that are ot solutios Ê y ; for Ÿ y Ÿ, f() g() yè y ay Î y y a y, ad y symmetry of the graph, A y yay dy c Î Î y a y c c Î a y dy y a y dy y ˆ () ˆ ˆ 8. AREA A A Limits of itegratio: y ad y y Ê y y y Ê y ay y y(y )(y ) Ê y,, : for Ÿ y Ÿ, f(y) g(y) y y y y y c y y a A Ê A ay y y dy y ˆ ; for Ÿ y Ÿ, f(y) g(y) y y y Ê A y y y dy y Ê ˆ ; Therefore, A 9. Limits of itegratio: y ad y Ê Ê Ê a ( )( ) Ê a ad ; f() g() c & Ê A a d ˆ ˆ ˆ 6. Limits of itegratio: y ad y Ê Ê a () Ê ( )( ) Ê a ad ; f() g() a c Ê A a d ˆ ˆ Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.
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