CHAPTER 5 INTEGRATION

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1 CHAPTER 5 INTEGRATION 5. AREA AND ESTIMATING WITH FINITE SUMS. fa Sice f is icreasig o Òß Ó, we use left edpoits to otai lower sums ad right edpoits to otai upper sums. i ) i i ( ( i ˆ i Š ˆ ˆ ˆ ) i i (a) ad i Ê a lower sum is ˆ i Š ˆ () ad i Ê a lower sum is & i ) i i & i ˆ i Š ˆ ˆ ˆ ˆ i i (c) ad i Ê a upper sum is ˆ i Š ˆ + (d) ad i Ê a upper sum is +. fa Sice f is decreasig o Òß 5 Ó, we use left edpoits to otai upper sums ad right edpoits to otai lower sums. & i i & & i & (( i ˆ i & i & ) i ˆ i i & & i ˆ i i (a) ad i i Ê a lower sum is ˆ () ad i i Ê a lower sum is (c) ad i i Ê a upper sum is (d) ad i i Ê a upper sum is Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.

2 Chapter 5 Itegratio 5. fa Usig rectagles Ê Ê ˆ fˆ fˆ & Š ˆ ˆ Usig rectagles Ê Ê ˆ & ( fˆ fˆ fˆ fˆ ) ) ) ) & ( Š ˆ ˆ ˆ ˆ ) ) ) ) 7. fa & Usig rectagles Ê Ê afa fa ˆ & Usig rectagles Ê Ê ˆ ˆ ˆ & ˆ ( f f f fˆ * ˆ )) * * & ( * & ( * & ( * & 9. (a) D ()() ()() ()() ()() (5)() ()() ()() (6)() ()() (6)() 87 iches () D ()() ()() ()() (5)() ()() ()() (6)() ()() (6)() ()() 87 iches. (a) D ()() ()() (5)() (5)() ()() ()() (5)() (5)() ()() (5)() ()() ()() 9 feet.66 miles () D ()() (5)() (5)() ()() ()() (5)() (5)() ()() (5)() ()() ()() (5)() 8 feet.7 miles. (a) Because the acceleratio is decreasig, a upper estimate is otaied usig left ed-poits i summig acceleratio? t. Thus,? t ad speed [ ]() 7.65 ft/sec () Usig right ed-poits we otai a lower estimate: speed [ ]() 5.8 ft/sec (c) Upper estimates for the speed at each secod are: t 5 v Thus, the distace falle whe t secods is s [ ]() 6.59 ft. 5. Partitio [ß ] ito the four suitervals [ß.5], [.5ß ], [ß.5], ad [.5ß ]. The midpoits of these suitervals are m.5, m.75, m.5, ad m.75. The heights of the four approimatig rectagles are f(m ) (.5), f(m ) (.75), f(m ) (.5), ad f(m ) (.75) ˆ ˆ ˆ ˆ ˆ 5 ˆ ˆ 7 ˆ Notice that the average value is approimated y approimate area uder legth of [ß]. We use this oservatio i solvig the et several eercises. curve f() 7. Partitio [ß] ito the four suitervals [ß.5], [.5ß], [ß.5], ad [.5ß]. The midpoits of the suitervals are m.5, m.75, m.5, ad m.75. The heights of the four approimatig rectagles are 5 È f(m ) si, f(m ) si, f(m ) si Š Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.

3 Sectio 5. Sigma Notatio ad Limits of Fiite Sums 7 È? ˆ area Ê legth of [ß ], ad f(m ) si Š. The width of each rectagle is. Thus, Area ( ) average value. 9. Sice the leaage is icreasig, a upper estimate uses right edpoits ad a lower estimate uses left edpoits: (a) upper estimate (7)() (97)() (6)() (9)() (65)() 758 gal, lower estimate (5)() (7)() (97)() (6)() (9)() 5 gal. () upper estimate ( ) 6 gal, lower estimate ( ) 69 gal. (c) worst case: 6 7t 5, Ê t. hrs; est case: 69 7t 5, Ê t. hrs. (a) The diagoal of the square has legth, so the side legth is È. Area Š È () Thi of the octago as a collectio of 6 right triagles with a hypoteuse of legth ad a acute agle measurig ). Area ˆ ˆ si ˆ cos si È Þ)) ) ) (c) Thi of the 6-go as a collectio of right triagles with a hypoteuse of legth ad a acute agle measurig. Area ˆ ˆ si ˆ cos ) si È Þ ) (d) Each area is less tha the area of the circle,. As icreases, the area approaches. 5. SIGMA NOTATION AND LIMITS OF FINITE SUMS 6 6() 6() cos cos( ) cos( ) cos( ) cos( ) È È 5. ( ) si ( ) si ( ) si ( ) si 6 7. (a) & 86 () 86 5 & (c) 86 All of them represet 86 c c c c 9. (a) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) () c (c) c ( ) ( ) ( ) ( ) (a) ad (c) are equivalet; () is ot equivalet to the other two. Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.

4 Chapter 5 Itegratio ( ) 7. (a) a a ( 5) () (6) (c) (a ) a 5 6 (d) (a ) a 5 6 (e) ( a ) a 6 ( 5) 6 ( ) 9. (a) 55 ( )(() ) 6 () 85 (c) ( ) (7 ) Š (6 )((6) ) a (6) ( 5) a 5 5 5(5 )((5) ) Š 5(5 ) 5 Š Œ Œ 5(5 ) 5(5 ) Š Š (a) a 7 5 () 7 7a (c) Let j Ê j ; if Ê j ad if 6 Ê j 6 Ê a6 6 j. (a) () c c (c) a a Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.

5 Sectio 5. Sigma Notatio ad Limits of Fiite Sums. (a) () (c) 5. (a) () (c) 7...,.5..,..5.8,.6.., ad.6.; the largest is lpl.. & 9. fa i Let ad c i. The right-had sum is i a c Š ˆ i i a i i i i a a i i. Thus, lim a c Ä_ i i lim Œ Ä_. fa i i c ˆ i * i i i i i ( ( a a i Š i Let ad c i. The right-had sum is a Š Š ( * * a ) ) lim lim Ä_ i i Ä_ Þ Thus, ( * ac Œ *. Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.

6 Chapter 5 Itegratio. fa a i i a i Š ˆ i i c ci i i i i i i Let ad c i. The right-had sum is a a a. Thus, lim aci c Ä_ i i Š Š lim Š Œ Ä_ &. 5. fa i Let ad c i. The right-had sum is i ˆ i a ci i i i i a Š Š a. lim a lim Ä_ i i Ä_ Thus, c. 5. THE DEFINITE INTEGRAL & (. d. a d cî 5. d 7. (sec ) d 9. (a) g() d () g() d g() d 8 & & & & (c) f() d f() d ( ) (d) f() d f() d f() d 6 ( ) & & & (e) [f() g()] d f() d g() d 6 8 & & & (f) [f() g()] d f() d g() d (6) 8 6. (a) f(u) du f() d 5 () È f(z) dz È f(z) dz 5È (c) f(t) dt f(t) dt 5 (d) [ f()] d f() d 5. (a) f(z) dz f(z) dz f(z) dz 7 () f(t) dt f(t) dt Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.

7 Sectio 5. The Defiite Itegral 5 5. The area of the trapezoid is A (B )h (5 )(6) Ê ˆ d square uits 7. The area of the semicircle is A r () 9 9 Ê È9 d square uits 9. The area of the triagle o the left is A h ()(). The area of the triagle o the right is A h ()(). The, the total area is.5 Ê d.5 square uits. The area of the triagular pea is A h ()(). The area of the rectagular ase is S jw ()(). The the total area is Ê a d square uits a. d ()( ) 5. s ds () a(a) a Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.

8 6 Chapter 5 Itegratio 7. (a) È d () a È d a È Š È () () 9. d. ) d) È 7 Š È 7 Î 7 ˆ. d 5. t dt a 7. d 9. d a. 7 d 7( ) È (a) Š È a a. (t ) dt t dt dt ( ) 6 z z 7 5. dz dz dz dz z dz [ ] ˆ ˆ 7. u du u du u du u du Š ˆ a 5 d d d 5 d 5[ ] (8 ) 5. Let? ad let,?,? ßá ß c ( )?,?. Let the c s e the right ed-poits of the suitervals Ê c, c, ad so o. The rectagles defied have areas: f(c )? f(? )? (? )? (? ) f(c )? f(? )? (? )? () (? ) f(c )? f(? )? (? )? () (? ) ã f(c )? f(? )? (? )? () (? ) The S f(c )? (? ) (? ) ()() Š Š 6 ˆ ˆ Ä_ Ê d lim. Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.

9 Sectio 5. The Defiite Itegral 7 5. Let? ad let,?,? ßá ß c ( )?,?. Let the c s e the right ed-poits of the suitervals Ê c, c, ad so o. The rectagles defied have areas: f(c )? f(? )? (? )(? ) (? ) f(c )? f(? )? (? )(? ) ()(? ) f(c )? f(? )? (? )(? ) ()(? ) ã f(c )? f(? )? (? )(? ) ()(? ) The S f(c )? (? )? ( Š Š ) Ä_ ( ) Ê d lim. È È È È È È 55. av(f) Š a d d d Š È È È Š È. 57. av(f) ˆ a d d d Š ( ). 59. av(f) ˆ (t ) dt t dt t dt dt Š Š ( ). Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.

10 8 Chapter 5 Itegratio ( ) ( ) d ( ) d 6. (a) av(g) Š a d d d d d ( ) Š ( ( )) Š ( ). () av(g) ˆ a d ( ) d Š d d ( ). ( ) (c) av(g) Š a d a d a d ( ) (see parts (a) ad () aove). 6. Cosider the partitio P that sudivides the iterval Òa, Ó ito suitervals of width a ad let c e the right Ö a aa aa aa a a c a a c a a a Ä_ m mä 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 65. Cosider the partitio P that sudivides the iterval Òa, Ó ito suitervals of width a ad let c e the right a aa aa aa Ö a ˆ Š a a a a a a a a aa Š a a a a a a a a a Œ a a a a a a a a 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. Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.

11 Sectio 5. The Defiite Itegral 9 a a a a Š 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. We get the Riema sum fac Š ˆ ˆ a aa aa a a a a a a a ˆ Š a a a a a a a a a a a a a a a a a a a a a a a a a aa a a a a a a a Š aaa a aa aaa a a aa a a a a a aa a a a Ä_ m mä a a a aa a a a aa aa a. Thus, d. a 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 Š Œ a a. As ad P this epressio has value 7. To fid where, let Ê ( ) Ê or. If, the Ê a ad maimize 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. 75. Ÿ si a Ÿ for all Ê ( )( ) Ÿ si a d Ÿ ( )() or si d Ÿ Ê si d caot equal. 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. a 79. si Ÿ for Ê si Ÿ for Ê (si ) d Ÿ (see Eercise 78) Ê si d d Ÿ si d d si d Š si d. Thus a upper oud is. Ê Ÿ Ê Ÿ Ê Ÿ 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 a Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.

12 Chapter 5 Itegratio 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 f() f(a)? ma sice f() f(a). Thus lim (U L) lim (f() f(a))? ma, sice? ma lp l. lpl Ä lpl Ä 85. (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 87. By Eercise 86, U L M m where M maö fa o the ith suiterval ad i i i i i i i m miö fa o the ith suiterval. Thus U L am m provided for each i i i i i i i i i ßÞÞÞ,. Sice a a the result, U L a a follows. i i i 5. THE FUNDAMENTAL THEOREM OF CALCULUS. ( 5) d c 5d a 5() a( ) 5( ) 6 c i. a d a d Š Š a a a a 5. Š d Š Š 8 () () () ˆ È d ˆ Î Î 9. sec d [ ta ] ˆ ta ˆ ( ta ) È È Î Î Î. csc ) cot ) d ) [ csc )] ˆ csc ˆ ˆ csc ˆ È Š È Î Î Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.

13 Sectio 5. The Fudametal Theorem of Calculus Î Î Î cos t. dt ˆ cos t dt t si t ˆ () si () ˆ ˆ si ˆ Î Î Î 5. ta d asec d [ta ] ˆ ta ˆ ata a 7. si d cos ˆ cos ˆ ˆ cos a Î8 Î8 È 8 r ( ) 8 9. (r ) dr ar r dr r r Š ( ) ( ) Š ) Š È u u & u u 6 È 6 6 Š È ( ( ) ). È Š & du È Š u du u Š () È È È. s Ès ds s ˆ Î s Î ds s È È Š È È ÈÈ 8 s ÉÈ si sicos 5. d d cos d si asi a ˆ si ˆ Î si Î si Î Î 7. d d d d d Š Š 6 c c c ( ) 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 Î d cos È È 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 dy. y t dt Ê È È d & dt È 5. y si t dt dy d Î È si t dt Ê si (si ) d Š ˆ È ˆ ˆ È ˆ d t t dy ct t d 7. y dt dt Ê si Èt d Èsi d Ècos cos cos dt dy d cos cos 9. y, Ê ˆ (si ) (cos ) sice d si È Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.

14 Chapter 5 Itegratio. Ê ( ) Ê or ; Area a d a d a d ( ) ( ) Š Š ( ) Š ( ) ( ) Š Š Š ( ) 8 Š Š Š. Ê a Ê ( )( ) Ê,, or ; Area a d a d Š Š Š Š 5. 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. 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. dy t d t 9. y dt Ê ad y( ) dt Ê (d) is a solutio to this prolem. dy d 5. y sec t dt Ê sec ad y() sec t dt Ê () is a solutio to this prolem. 5. y sec t dt Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.

15 Î Î 55. 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 Sectio 5.5 Idefiite Itegrals ad the Sustitutio Rule cî h dc 57. Î Ê c Î t dt Î t È; c() c() È È 9. d È 59. (a) t Ê T 85 È 5 7 F; t 6 Ê T 85 È F; È t 5 Ê T F 5 Î () average temperatuve Š 85 È5 t dt 85t a5 t 5 5 Î Î 5 5 Š 85a5a5 5 Š 85aa5 75 F d d 6. f(t) dt Ê f() d f(t) dt d a 5 9 w 9 9 w 9 t () t 6. f() dt Ê f () Ê f () ; f() dt ; L() ( ) f() ( ) (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 () ). 5.5 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 5 Ê du d Ê du d a a 5 d u du u du u C 5 C 5. Let u Ê du a6 d a d Ê du a d 5 a a a d u du u du u C C 7. Let u Ê du d Ê du d si d si u du cos u C cos C 5 Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.

16 Chapter 5 Itegratio 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. 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 Î Î. Let u Ê du d Ê du È d È si ˆ Î d si u du ˆ u si u C ˆ Î si ˆ Î C 6 u u Š 5. (a) Let u cot ) Ê du csc ) d ) Ê du csc ) d) csc ) cot ) d) u du C C cot ) C u u Š () Let u csc ) Ê du csc ) cot ) d ) Ê du csc ) cot ) d) csc ) cot ) d) u du C C csc ) C 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 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 È È È du Ȉ È u u È. Let u Ê du d Ê du d d C C 5 5. Let u Ê du d Ê du d sec ( ) d asec u ˆ du sec u du ta u C ta ( ) C 5. 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 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 Î Î Î 9. Let u Ê du d Ê du d Î si ˆ Î d (si u) ˆ du Î si u du ( cos u) C cos ˆ 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 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 t t t Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.

17 5. Let u si Ê du ˆ cos ˆ d ) Ê du cos d) ) ) ) ) ) ) ) ) ) si cos d) u du u C si C 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 9. Let u Ê du d Î Î Î É d Èu du u du u C ˆ C Sectio 5.5 Idefiite Itegrals ad the Sustitutio Rule 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 Let u. The du d ad u. Thus a d au u du au u du u u C a a C 5 5. 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 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 a u C a C 9. Let u Ê du d ad du d. Thus d d u du u du a 5. (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 5. 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 Î Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.

18 6 Chapter 5 Itegratio 55. 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 5 Ê s t Let u t Ê du dt 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 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 5) du si u 5u C si ˆ t 5 ˆ t C ; ˆ ˆ at t ad s we have si 5 C Ê C 5 Ê s si ˆ t t 5 ( 5 ) Ê s si ˆ t t 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. 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. 5.6 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 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 t t dt u 5 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 Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.

19 5 5r dr 5 u du a r 5 Sectio 5.6 Sustitutio ad Area Betwee Curves 7 7. (a) Let u r Ê du r dr Ê du r dr; r Ê u 5, r Ê u 5 () Use the same sustitutio as i part (a); r Ê u, r Ê u 5 5 5r & dr 5 u du 5 u 5 (5) 5 () a r ˆ ˆ 8 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 6 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. (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 & 5. Let u t t Ê du a5t dt; t Ê u, t Ê u Èt& t a5t dt Î u du Î u Î Î () () È 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 () 9. Let u 5 cos t Ê du si t dt Ê du si t dt; t Ê u 5 cos, t Ê u 5 cos * 5 Î Î 5 (5 cos t) si t dt 5u ˆ du 5 Î u du 5 ˆ &Î &Î &Î u 9. Let u y y y Ê du a y y dy; y Ê u, y Ê u () () () 8 8 Î y y y y y dy Î a a u du Î ) u Î Î (8) () 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 () Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.

20 8 Chapter 5 Itegratio È È Î Î Î Î c Î Î Î 6 u () () 5. 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 7. 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 () () cos 9. For the setch give, a, ; f() g() cos si ; (cos ) A d ( cos ) d si [( ) ( )]. For the setch give, a, ; f() g() a ; & A a d ˆ ˆ c For the setch 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 () () 5. 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 AREA A A A: For the setch 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 ˆ ( 8 9 ) 9 ; A: For the setch 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 A: For the setch give, a ad : f() g() ( ) a c Ê A a d ˆ ˆ 8 7 ; c 6 6 A: For the setch give, a ad : f() g() a ( ) a Ê A a d ˆ 8 ˆ 9 8 ; c A: For the setch give, a ad : f() g() ( ) a Ê A a d ˆ ˆ ; 9 ˆ Therefore, AREA A A A 9 9 Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.

21 . a, ; f() g() a Ê A a d ˆ 8 8 ˆ 8 8 c 8 ˆ Sectio 5.6 Sustitutio ad Area Betwee Curves 9. a, ; f() g() 8 Ê A a8 d & 5. Limits of itegratio: Ê Ê ( ) Ê a ad ; f() g() a Ê A a d 7. Limits of itegratio: Ê 5 Ê a a Ê ( )( )( )( ) Ê,,, ; f() g() a 5 ad g() f() a 5 c c c Ê A a 5 d a 5 d a 5 d & & & ˆ ˆ 8 ˆ ˆ ˆ 8 ˆ 8 È, Ÿ 9. Limits of itegratio: y È ad È, 6 5y 6 or y ; for Ÿ : È Ê 5È 6 Ê 5( ) 6 Ê 7 6 Ê ( )( 6) Ê, 6 (ut 6 is ot a solutio); for : 5È 6 Ê 5 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 Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.

22 5 Chapter 5 Itegratio ( 6) Î ( 6) Î Î ( 6) * 6 5 Î 6 Î 5 Î ( ) ˆ ˆ ˆ 5. Limits of itegratio: c ad d ; f(y) g(y) y y Ê A y dy 9 8 y 5. Limits of itegratio: y ad 6 y Ê y 6 y Ê y y Ê (y 5)(y ) Ê c ad d 5; f(y) g(y) ˆ Š 6y y y y 5 c y y & ˆ Ê A ay y dy y ˆ 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 57. Limits of itegratio: y ad yè y Ê y yèy 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 Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.

23 59. Limits of itegratio: y ad y Ê Ê 5 Ê a 5 ( )( ) Ê a ad ; f() g() 5 c & Ê A a 5 d 5 ˆ ˆ ˆ 6. Limits of itegratio: y ad y Ê y y Ê y y Ê Š y È Š y È (y )(y ) Ê c ad d sice ; f(y) g(y) a y a y y y Ê A a y y dy y y y ˆ c & 6. a, ; f() g() si si Ê A ( si si ) d cos cos ( ) ˆ Sectio 5.6 Sustitutio ad Area Betwee Curves a, ; f() g() a cos ˆ Ê A cos ˆ d si ˆ c ˆ ˆ ˆ Î 67. a, ; f() g() sec ta Ê A asec ta d Î cî csec asec d d cî Î d [] ˆ cî Î Î 69. c, d ; f(y) g(y) si yècos y si yè cos y Ê A si yècos y dy (cos y) Î ( ) Î Î Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.

24 5 Chapter 5 Itegratio 7. A A A Limits of itegratio: y ad y Ê y y Ê y y Ê y(y )(y ) Ê c, d ad c, d ; f (y) g (y) y y ad f (y) g (y) y y Ê y symmetry aout the origi, A A A Ê A ay y dy ˆ y y 7. A A A Limits of itegratio: y ad y Ê, Á Ê Ê, f () g () Ê A d ; f () g () Ê A d ; A A A 75. (a) The coordiates of the poits of itersectio of the lie ad paraola are c Ê Èc ad y c () f(y) g(y) Èy ˆ Èy Èy Ê the area of the lower sectio is, A [f(y) g(y)] dy c L c c Èy dy Î y Î c. The area of the etire shaded regio ca e foud y settig c : A ˆ Î 8. Sice we wat c to divide the regio ito susectios of equal area we have A A Ê ˆ Î L c Î Ê c Èc Èc Èc Î (c) f() g() c Ê A [f() g()] d Î c L È c d c c c È a c È c c c c Î L Î Ê Î c. Agai, the area of the whole shaded regio ca e foud y settig c Ê A. From the coditio A A, we get c c as i part (). 77. Limits of itegratio: y È ad y È Ê È, Á Ê È Ê ( ) È Ê Ê 5 Ê ( )( ) Ê, (ut does ot satisfy the equatio); y ad y Ê È È Ê 8 È Ê 6 Ê. Therefore, AREA A A : f () g () ˆ Î Ê A ˆ Î Î d ˆ 7 Î ; f () g () Ê A ˆ Î Î d ˆ ˆ ; Therefore, AREA A A 79. Area etwee paraola ad y a : A aa d a Š a ; a a a a a a Ä a Š Area of triagle AOC: (a) aa a ; limit of ratio lim which is idepedet of a. Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.

25 Chapter 5 Practice Eercises 5 8. The lower oudary of the regio is the lie through the poits azß z ad Š z ß az. The equatio of this ˆ z ˆ a z zz z lie is yaz a aza Ê y azaz z. The area of theregio is give y aa aaz az z dy z z a az z zdy az az z z Š az az az az zaz ˆ z az z az zz. No matter where we 6 choose z, the area of the regio ouded y y ad the lie through the poits azß z ad Š z ß az is always 6. z z 6 6 si si u si u ˆ u u 8. Let u Ê du d Ê du d; Ê u, Ê u 6 d ˆ du du cf(u) d F(6) F() 85. (a) Let u Ê du d; Ê u, Ê u f odd Ê f( ) f(). The f() d f( u) ( du) f(u) ( du) f(u) du f(u) du () Let u Ê du d; Ê u, Ê u c f eve Ê f( ) f(). The f() d f( u) ( du) f(u) du f(u) du c 87. Let u a Ê du d; Ê u a, a Ê u a a a a f() d f(a) d f() f(a) f() d f(au) f(au) du f(a) d f() f(a) f(au) f(u) f(u) f(au) f() f(a) a a a a I ( du) a Ê I I d d [] a a. f() f(a) f() f(a) f() f(a) Therefore, I a Ê I a. 89. Let u c Ê du d; a c Ê u a, c Ê u c c f( c) d f(u) du f() d acc a a CHAPTER 5 PRACTICE EXERCISES. (a) Each time suiterval is of legth? t. sec. The distace traveled over each suiterval, usig the midpoit rule, is? h a v v i i? t, where v i is the velocity at the left edpoit ad v i the velocity at the right edpoit of the suiterval. We the add? h to the height attaied so far at the left edpoit v i to arrive at the height associated with velocity v i at the right edpoit. Usig this methodology we uild the followig tale ased o the figure i the tet: t (sec) v (fps) h (ft) t (sec) v (fps) h (ft) Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.

26 5 Chapter 5 Itegratio NOTE: Your tale values may vary slightly from ours depedig o the v-values you read from the graph. Rememer that some shiftig of the graph occurs i the pritig process. The total height attaied is aout 68 ft. () The graph is ased o the tale i part (a).. (a) a a ( ) () ( a ) a 5 ( ) (c) (a ) a 5 ()() (d) ˆ () 5 5. Let u Ê du d Ê du d; Ê u, 5 Ê u Î Î ( ) d u ˆ du Î u 7. Let u Ê du d; Ê u, Ê u cos ˆ d (cos u)( du) [ si u] si si ˆ ( ( )) c cî Î * c c c c f() g() (a) f() d f() d () () f() d f() d f() d 6 c c c c (c) g() d g() d (d) ( g()) d g() d () (e) Š d f() d g() d (6) () c 5 5 c 5 c Ê ( )( ) Ê or ; Area a d a d Š () () 8 Š () () Š () () ˆ ˆ Î Î. 5 5 Ê Ê ; Area ˆ Î 55 d ˆ Î 55 d 8 c &Î &Î ) &Î 5 5 ˆ 5() () ˆ &Î 5( ) ( ) ˆ &Î 5(8) (8) ˆ &Î 5() () [ ( )] [( 96) ] 6 Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.

27 Chapter 5 Practice Eercises f(), g(), a, Ê A [f() g()] d a ˆ d ˆ ˆ 7. f() ˆ È, g(), a, Ê A [f() g()] d ˆ È d ˆ È d ˆ Î Î d (68) 9. f(y) y, g(y), c, d Ê A [f(y) g(y)] dy ay dy d c y dy c y d 8 a 6 6. Let us fid the itersectio poits: y y Ê y y Ê (y )(y ) Ê y y y d A [f(y) g(y)] dy Š y y dy c c y y dy y y a y c ˆ 8 ˆ 9 8 or y Ê c, d ; f(y), g(y) Ê. f(), g() si, a, Ê A [f() g()] d ( si ) d Î a Î È cos Š 5. a,, f() g() si si Ê A ( si si ) d cos cos ( ) ˆ Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.

28 56 Chapter 5 Itegratio 7. f(y) Èy, g(y) y, c, d d Ê A [f(y) g(y)] dy È y ( y) dy c ˆ È y y dy Î y y y 8È7 6 6 Š È ˆ È 7 w w 9. f() ( ) Ê f () 6 ( ) Ê f ± ± Ê f() is a maimum ad f() is a miimum. A a d ˆ 7 Î. The area aove the -ais is A ˆ y y dy &Î y y 5 ; the area elow the -ais is A Î y y dy &Î ˆ y y c Ê the total area is A A dy d y w t d d t. y dt Ê Ê ; y() dt ad y () 5 si t dy si si t 5 t d 5 t 5. y dt Ê ; 5 Ê y dt 7. Let u cos Ê du si d Ê du si d Î Î Î Î Î (cos ) si d u ( du) u du Š C u C (cos ) C 9. Let u ) Ê du d ) Ê du d) [) cos () )] d ) (u cos u) ˆ du u () ) si u C si () ) C ) ) si () ) C, where C C is still a aritrary costat u Î c. ˆ t ˆ t dt ˆ t t t t dt at t dt Š C C t t t t Î. Let u t Ê du È t dt Ê du Èt dt Èt si ˆ Î t dt si u du cos u C cosˆ Î t C 5. a 7 d c 7d c () 7() dc( ) ( ) 7( ) d 6 ( ) 6 c v 7. dv v dv cv d ˆ ˆ Î Î tèt tî È È dt dt ( ) 9. t dt t Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.

29 5. Let u Ê du d Ê 8 du 6 d; Ê u, Ê u 6 d 8u () u c 8u du ˆ ˆ 8 Chapter 5 Practice Eercises 57 Î Î Î 5. Let u Ê du d Ê du d; Ê u ˆ Î, Ê u Î 8 8 Î Î Î &Î d Î &Î &Î &Î ˆ u ˆ du ˆ u u () ˆ ˆ Î8 Î Š 5 Î 5 Î È Let u 5r Ê du 5 dr Ê 5 du dr; r Ê u, r Ê u 5 5 si 5r dr & asi u ˆ du u si u ˆ si ˆ si 5 5 Î 57. sec ) d ) [ta )] ta ta È Î Î Î Î cot d 6 cot u du 6 csc u du [6( cot u u)] 6 cot 6 cot Î 6 6 Î Î 59. Let u Ê du d Ê 6 du d; Ê u, Ê u 6È 6. sec ta d [sec ] sec sec ˆ cî Î a ˆ ˆ 6. Let u si Ê du cos d; Ê u, Ê u Î Î Î &Î &Î &Î &Î 5(si ) cos d 5u du 5 ˆ u u () () Let u si Ê du cos d Ê du cos d; Ê u si ˆ, Ê u si ˆ Î c c & & & 5 si cos d 5u ˆ du 5u du u ( ) () c d c Î Î Î si cos Î u Î Î Î d du u du u È ˆ si Èu Š 67. Let u si Ê du 6 si cos d Ê du si cos d; Ê u, Ê u si 69. Let u sec ) Ê du sec ) ta ) d ); ) Ê u sec, ) Ê u sec Î Î Î ta ) sec ) ta ) sec ) ta ) Î È ) sec ) sec ) È ) sec ) È ) (sec )) È Î uî È u È cî ˆ È u È Š () È È () d d d du u du m m() m( ) ( ) c m m() m( ) ( ) c c Š Š 7. (a) av(f) (m ) d Š () Š ( ) () () av(f) (m ) d () ( ) () w w f() f(a) w av a a a a a a 7. f f () d [f()] [f() f(a)] so the average value of f over [aß] is the slope of the secat lie joiig the poits (aßf(a)) ad (ßf()), which is the average rate of chage of f over [aß]. Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.

30 58 Chapter 5 Itegratio 75. We wat to evaluate & & & & ( & & & & & & & f() d Œ( si a & d si a d d & Notice that the period of y si a is & ad that we are itegratig this fuctio over a iterval of & ( & ( & legth 65. Thus the value of si a d d is & &. & & & & & & & dy d È 77. cos dy d d d t 79. Œ dt 8. Yes. The fuctio f, eig differetiale o [aß], is the cotiuous o [aß]. The Fudametal Theorem of Calculus says that every cotiuous fuctio o [aß] is the derivative of a fuctio o [aß]. d d d 8. y È t dt È dy d d t dt Ê È t dt È t dt È 85. We estimate the area A usig midpoits of the vertical itervals, ad we will estimate the width of the parig lot o each iterval y averagig the widths at top ad ottom. This gives the estimate A & ˆ & && &*Þ& *Þ&& &Þ Þ(Þ& (Þ& A &* ft. The cost is Area (./ft ) a596 ft a./ft,58. Ê the jo caot e doe for,. CHAPTER 5 ADDITIONAL AND ADVANCED EXERCISES. (a) Yes, ecause f() d 7f() d (7) 7 7 Î c d È È È Š ˆ Î Î È () No. For eample, 8 d, ut 8 d Á È a a a si a cos a dy a a d Œ. y f(t) si a( t) dt f(t) si a cos at dt f(t) cos a si at dt f(t) cos at dt f(t) si at dt Ê cos a f(t) cos at dt si a d cos a d a d a d Œ f(t) cos at dtsi a f(t) si at dt Œ f(t) si at dt si a a cos a f(t) cos at dt (f() cos a) si a f(t) si at dt (f() si a) Ê cos a f(t) cos at dt si a f(t) si at dt. Net, dy d cos a a dy d d a si a f(t) cos at dt (cos a) Œ f(t) cos at dta cos a f(t) si at dt d d d (si a) Œ f(t) si at dt a si a f(t) cos at dt (cos a)f() cos a ww a cos a f(t) si at dt (si a)f() si a a si a f(t) cos at dt a cos a f(t) si at dt f(). Therefore, y a y a cos a f(t) si at dt a si a f(t) cos at dt f() si a cos a w Œ a a a f(t) cos at dt f(t) si at dt f(). Note also that y () y(). Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.

31 d Chapter 5 Additioal ad Advaced Eercises (a) f(t) dt cos Ê d f(t) dt cos si Ê f a () cos si cos si cos si f a. Thus, f() f() f() t Ê Ê () t dt (f()) Ê (f()) cos Ê (f()) cos Ê f() È cos Ê f() È () cos È d d È È 7. f() d È È Î Ê f() f() d a () Ê f() dy 9. d Ê y a d C. The (ß) o the curve Ê () C Ê C Ê y Î. f() d d d c8 c8 &Î 5 [ ] ) ˆ &Î ( 8) ( () ). g(t) dt t dt si t dt t cos t ˆ cos ˆ cos c 5. f() d d a d d c c c [] [] ( ) a ( ) Š Š () () ˆ a a 7. Ave. value f() d f() d d ( ) d Š Š Š &? _ & ß j Š ˆ j 9. Let f() o [ß]. Partitio [ß] ito suitervals with. The,, á, are the right-had edpoits of the suitervals. Sice f is icreasig o [ ], U is the upper sum for _ & & & & & f() o [ß] Ê lim j á Š ˆ lim ˆ ˆ á ˆ lim & & & Ä_ Ä_ Ä_ & 6 6 d j Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.

32 6 Chapter 5 Itegratio? _ ß j Š ˆ j j ˆ ˆ ˆ Ä_. Let y f() o [ß]. Partitio [ß] ito suitervals with. The,, á, are the right-had edpoits of the suitervals. Sice f is cotiuous o [ ], f is a Riema sum of _ y f() o [ß] Ê lim f Š ˆ lim f f á f f() d Ä_ j. (a) Let the polygo e iscried i a circle of radius r. If we draw a radius from the ceter of the circle (ad the polygo) to each verte of the polygo, we have isosceles triagles formed (the equal sides are equal to r, the radius of the circle) ad a verte agle of ) where ). The area of each triagle is r r A r si ) Ê the area of the polygo is A A si ) si. Ä_ Ä_ r r Ä_ si Ä_ si a ˆ a ˆ ˆ ˆ Î Ä () lim A lim si lim si lim r r lim r 5. (a) ga fat dt () ga fat dt a a (c) ga fa t dt fa t dt a w w (d) g a fa Ê,, ad the sig chart for g a fa is ± ± ±. So g has a relative maimum at. w (e) g a f a is the slope ad ga fat dt, y (c). Thus the equatio is y a y. (f) g ww a f w a at ad g ww a f w a is egative o aß ad positive o aß so there is a ww w ww w iflectio poit for g at. We otice that g a f a for o aß ad g a f a for o ww aß, eve though g a does ot eist, g has a taget lie at, so there is a iflectio poit at. (g) g is cotiuous o Òß Ó ad so it attais its asolute maimum ad miimum values o this iterval. We saw i (d) that g w a Ê,,. We have that ga fa t dt fa t dt ga fat dt ga fat dt ga fat dt Thus, the asolute miimum is ad the asolute maimum is. Thus, the rage is Ò ß Ó. d d / t d d 7. f() dt Ê f w () ˆ Š ˆ ˆ ˆ Èy 9. g(y) w È si t dt Ê g (y) Š si ˆ y d Š ˆ y Š si ˆ y d Š ˆ y si y È È È È y si y dy dy Èy Èy Copyright Pearso Educatio, Ic. Pulishig as Addiso-Wesley.