CHAPTER 5 INTEGRATION

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CHAPTER 5 INTEGRATION 5.1 AREA AND ESTIMATING WITH FINITE SUMS 1. fax x Sce f s creasg o Ò!ß Ó, we use left edpots to ota lower sums ad rght edpots to ota upper sums.! )!! ( (!ˆ 4 4 4Š ˆ ˆ ˆ 4 4 )! (a) x ad x x Ê a lower sum s!ˆ Š! ˆ () x ad x x Ê a lower sum s!! & ) 1!! &!ˆ 4 4 4Š ˆ ˆ ˆ 4 4 ˆ (c) x ad x x Ê a upper sum s!ˆ Š ˆ +1 (d) x ad x x Ê a upper sum s +1. fax x Sce f s creasg o Ò!ß Ó, we use left edpots to ota lower sums ad rght edpots to ota upper sums.!!! *!ˆ 4 4 4Š ˆ ˆ ˆ 4 4 &! (a) x ad x x Ê a lower sum s!ˆ Š! ˆ () x ad x x Ê a lower sum s!! * * ) 1!!! &!ˆ 4 4 4Š ˆ ˆ ˆ 4 4 & (c) x ad x x Ê a upper sum s!ˆ Š ˆ +1 (d) x ad x x Ê a upper sum s +1 Copyrght 010 Pearso Educato, Ic. Pulshg as Addso-Wesley.

58 Chapter 5 Itegrato 3. fax x Sce f s decreasg o Ò1ß 5 Ó, we use left edpots to ota upper sums ad rght edpots to ota lower sums. & x & & & ((! ˆ x &! & )! ˆ x! & &! ˆ x! (a) x ad x x Ê a lower sum s! ˆ () x 1 ad x x Ê a lower sum s (c) x ad x x Ê a upper sum s (d) x 1 ad x x Ê a upper sum s 4. fax x Sce f s creasg o Ò ß!Ó ad decreasg o Ò!ß Ó, we use left edpots o Ò ß!Ó ad rght edpots o Ò!ß Ó to ota lower sums ad use rght edpots o Ò ß!Ó ad left edpots o Ò!ß Ó to ota upper sums. a (a) x ad x x Ê a lower sum s ˆ a a! a () x ad x x Ê a lower sum s! ˆ ax! ˆ ax! ˆ ˆ a ˆ a a a a (c) x ad x x Ê a upper sum s ˆ a! a! a (d) x ad x x Ê a upper sum s! ˆ ax! ˆ ax ˆ ˆ a a! a! a 5. fax x! Usg rectagles Ê x Ê ˆ fˆ fˆ! & Š ˆ ˆ! Usg 4 rectagles Ê x Ê ˆ & ( fˆ fˆ fˆ fˆ ) ) ) ) & ( Š ˆ ˆ ˆ ˆ ) ) ) ) Copyrght 010 Pearso Educato, Ic. Pulshg as Addso-Wesley.

Secto 5.1 Area ad Estmatg wth Fte Sums 59 6. fax x! Usg rectagles Ê x Ê ˆ fˆ fˆ ) ( Š ˆ ˆ! Usg 4 rectagles Ê x Ê ˆ & ( fˆ fˆ fˆ fˆ ) ) ) ) & ( * ) ) ) ) Š 7. fax x & Usg rectagles Ê x Ê afa fa ˆ & Usg 4 rectagles Ê x Ê ˆ ˆ ˆ & ˆ ( f f f fˆ * ˆ )) * * & ( * & ( * & ( * & 8. fax x a Usg rectagles Ê x Ê afa fa a a Usg 4 rectagles Ê x Ê ˆ ˆ ˆ ˆ f f f fˆ ˆ ˆ ˆ ˆ *! Š Š Š Š Š ˆ 9. (a) D (0)(1) (1)(1) ()(1) (10)(1) (5)(1) (13)(1) (11)(1) (6)(1) ()(1) (6)(1) 87 ches () D (1)(1) ()(1) (10)(1) (5)(1) (13)(1) (11)(1) (6)(1) ()(1) (6)(1) (0)(1) 87 ches 10. (a) D (1)(300) (1.)(300) (1.7)(300) (.0)(300) (1.8)(300) (1.6)(300) (1.4)(300) (1.)(300) (1.0)(300) (1.8)(300) (1.5)(300) (1.)(300) 50 meters (NOTE: 5 mutes 300 secods) () D (1.)(300) (1.7)(300) (.0)(300) (1.8)(300) (1.6)(300) (1.4)(300) (1.)(300) (1.0)(300) (1.8)(300) (1.5)(300) (1.)(300) (0)(300) 490 meters (NOTE: 5 mutes 300 secods) 11. (a) D (0)(10) (44)(10) (15)(10) (35)(10) (30)(10) (44)(10) (35)(10) (15)(10) ()(10) (35)(10) (44)(10) (30)(10) 3490 feet 0.66 mles () D (44)(10) (15)(10) (35)(10) (30)(10) (44)(10) (35)(10) (15)(10) ()(10) (35)(10) (44)(10) (30)(10) (35)(10) 3840 feet 0.73 mles 1. (a) The dstace traveled wll e the area uder the curve. We wll use the approxmate veloctes at the mdpots of each tme terval to approxmate ths area usg rectagles. Thus, D (0)(0.001) (50)(0.001) (7)(0.001) (90)(0.001) (10)(0.001) (11)(0.001) (10)(0.001) (18)(0.001) (134)(0.001) (139)(0.001) 0.967 mles () Roughly, after 0.0063 hours, the car would have goe 0.484 mles, where 0.0060 hours.7 sec. At.7 sec, the velocty was approxmately 10 m/hr. Copyrght 010 Pearso Educato, Ic. Pulshg as Addso-Wesley.

60 Chapter 5 Itegrato 13. (a) Because the accelerato s decreasg, a upper estmate s otaed usg left ed-pots summg accelerato? t. Thus,? t 1 ad speed [3.00 19.41 11.77 7.14 4.33](1) 74.65 ft/sec () Usg rght ed-pots we ota a lower estmate: speed [19.41 11.77 7.14 4.33.63](1) 45.8 ft/sec (c) Upper estmates for the speed at each secod are: t 0 1 3 4 5 v 0 3.00 51.41 63.18 70.3 74.65 Thus, the dstace falle whe t 3 secods s s [3.00 51.41 63.18](1) 146.59 ft. 14. (a) The speed s a decreasg fucto of tme Ê rght ed-pots gve a lower estmate for the heght (dstace) attaed. Also t 0 1 3 4 5 v 400 368 336 304 7 40 gves the tme-velocty tale y sutractg the costat g 3 from the speed at each tme cremet? t 1 sec. Thus, the speed 40 ft/sec after 5 secods. () A lower estmate for heght attaed s h [368 336 304 7 40](1) 150 ft. 15. Partto [!ß ] to the four sutervals [0ß 0.5], [0.5ß 1], [1ß 1.5], ad [1.5ß ]. The mdpots of these sutervals are m 0.5, m 0.75, m 1.5, ad m 1.75. The heghts of the four approxmatg rectagles are f(m ) (0.5), f(m ) (0.75), f(m ) (1.5), ad f(m ) (1.75) 1 7 15 343 64 64 64 64 ˆ ˆ ˆ 3 ˆ ˆ 5 ˆ ˆ 7 ˆ 4 4 4 4 Notce that the average value s approxmated y approxmate area uder legth of [!ß]. We use ths oservato solvg the ext several exercses. curve f(x) x 16. Partto [1ß 9] to the four sutervals [ß], [3 ß& ], [&ß(], ad [(ß*]. The mdpots of these sutervals are m, m 4, m 6, ad m 8. The heghts of the four approxmatg rectagles are f(m ), f(m ), f(m ), ad f(m ). The wdth of each rectagle s x. Thus, 4 6 8? ˆ 5 4 6 8 1 legth of [ß*] 8 96 Area ˆ ˆ ˆ ˆ 5 area 1 5 Ê average value. 17. Partto [0ß] to the four sutervals [0ß0.5], [0.5ß1], [1ß1.5], ad [1.5ß]. The mdpots of the sutervals are m 0.5, m 0.75, m 1.5, ad m 1.75. The heghts of the four approxmatg rectagles are 1 31 51 4 4 4 È 71 4 Š È? ˆ area Ê legth of [0ß ] f(m ) s 1, f(m ) s 1, f(m ) s Š 1, ad f(m ) s 1. The wdth of each rectagle s x. Thus, Area (1 1 1 1) average value 1. 18. Partto [0ß4] to the four sutervals [0ß1], [1ß ß], [ß3], ad [3ß4]. The mdpots of the sutervals 3 5 7 are m, m, m, ad m. The heghts of the four approxmatg rectagles are 1 ˆ 1 Š 4 8 f(m ) 1 cos Š 1 ˆ cos ˆ 0.7145 (to 5 decmal places), f(m ) 1 Š cos Š 1 ˆ cos ˆ 31 0.97855, f(m ) 1 Š cos Š 1 ˆ cos ˆ 51 1 ˆ 3 1 ˆ 5 4 8 3 4 8 0.97855, ad f(m ) 1 Š cos Š 1 ˆ cos ˆ 7 0.7145. The wdth of each rectagle s 1 ˆ 7 1 4 8? x. Thus, Area (0.7145)(1) (0.97855)(1) (0.97855)(1) (0.7145)(1).5 Ê average area.5 5 legth of [0ß4] 4 8 value. Copyrght 010 Pearso Educato, Ic. Pulshg as Addso-Wesley.

Secto 5.1 Area ad Estmatg wth Fte Sums 61 19. Sce the leaage s creasg, a upper estmate uses rght edpots ad a lower estmate uses left edpots: (a) upper estmate (70)(1) (97)(1) (136)(1) (190)(1) (65)(1) 758 gal, lower estmate (50)(1) (70)(1) (97)(1) (136)(1) (190)(1) 543 gal. () upper estmate (70 97 136 190 65 369 516 70) 363 gal, lower estmate (50 70 97 136 190 65 369 516) 1693 gal. (c) worst case: 363 70t 5,000 Ê t 31.4 hrs; est case: 1693 70t 5,000 Ê t 3.4 hrs 0. Sce the pollutat release creases over tme, a upper estmate uses rght edpots ad a lower estmate uses left edpots: (a) upper estmate (0.)(30) (0.5)(30) (0.7)(30) (0.34)(30) (0.45)(30) (0.5)(30) 60.9 tos lower estmate (0.05)(30) (0.)(30) (0.5)(30) (0.7)(30) (0.34)(30) (0.45)(30) 46.8 tos () Usg the lower (est case) estmate: 46.8 (0.5)(30) (0.63)(30) (0.70)(30) (0.81)(30) 16.6 tos, so ear the ed of Septemer 15 tos of pollutats wll have ee released. 1. (a) The dagoal of the square has legth, so the sde legth s È. Area Š È () Th of the octago as a collecto of 16 rght tragles wth a hypoteuse of legth 1 ad a acute agle measurg 1 1 ). Area ˆ ˆ 1 s ˆ 1 cos 1 s È Þ)) ) ) (c) Th of the 16-go as a collecto of 3 rght tragles wth a hypoteuse of legth 1 ad a acute agle measurg 1 1. Area ˆ ˆ 1 s ˆ 1 cos 1 ) s È Þ! ) (d) Each area s less tha the area of the crcle, 1. As creases, the area approaches 1.. (a) Each of the sosceles tragles s made up of two rght tragles havg hypoteuse 1 ad a acute agle measurg 1 1. The area of each sosceles tragle s A ˆ ˆ s 1 ˆ cos 1 s 1. T 1 1 1 s P T 1 Ä_ Ä_ ˆ () The area of the polygo s A A s, so lm s lm 1 1 (c) Multply each area y r. A r s 1 T 1 AP r s lm AP 1r Ä_ 3-6. Example CAS commads: Maple: wth( Studet[Calculus1] ); f := x -> s(x); a := 0; := P; plot( f(x), x=a.., ttle=3(a) (Secto 5.1) ); N := [ 100, 00, 1000 ]; for N do Xlst := [ a+1.*(-a)/* =0.. ]; Ylst := map( f, Xlst ); ed do: for N do () (c) Copyrght 010 Pearso Educato, Ic. Pulshg as Addso-Wesley.

6 Chapter 5 Itegrato Avg[] := evalf(add(y,y=ylst)/ops(ylst)); ed do; avg := FuctoAverage( f(x), x=a.., output=value ); evalf( avg ); FuctoAverage(f(x),x=a..,output=plot); (d) fsolve( f(x)=avg, x=0.5 ); fsolve( f(x)=avg, x=.5 ); fsolve( f(x)=avg[1000], x=0.5 ); fsolve( f(x)=avg[1000], x=.5 ); Mathematca: (assged fucto ad values for a ad may vary): Symols for 1, Ä, powers, roots, fractos, etc. are avalale Palettes (uder Fle). Never sert a space etwee the ame of a fucto ad ts argumet. Clear[x] f[x_]:=x S[1/x] {a,}={ 1/4, 1} Plot[f[x],{x, a, }] The followg code computes the value of the fucto for each terval mdpot ad the fds the average. Each sequece of commads for a dfferet value of (umer of sudvsos) should e placed a separate cell. =100; dx = ( a) /; values = Tale[N[f[x]], {x, a dx/,, dx}] average=sum[values[[]],{, 1, Legth[values]}] / =00; dx = ( a) /; values = Tale[N[f[x]],{x, a + dx/,, dx}] average=sum[values[[]],{, 1, Legth[values]}] / =1000; dx = ( a) /; values = Tale[N[f[x]],{x, a dx/,, dx}] average=sum[values[[]],{, 1, Legth[values]}] / FdRoot[f[x] == average,{x, a}] 5. SIGMA NOTATION AND LIMITS OF FINITE SUMS! 1 6 6(1) 6() 6 1 1 1 1 1 3 1. 7! 3 1 1 1 1 3 1 1 7 1 3 3 6 1. 0! 4 1 3. cos 1 cos(1 1) cos( 1) cos(3 1) cos(4 1) 1 1 1 1 0! 5 1 4. s 1 s(1 1) s( 1) s(3 1) s(4 1) s(5 1) 0 0 0 0 0 0! 3 1 1 1 1 1 È3 È3 1 3 1 5. ( 1) s ( 1) s ( 1) s ( ) s 0 1! 4 1 6. ( 1) cos 1 ( 1) cos (11) ( 1) cos (1) ( 1) cos (31) ( 1) cos (41) ( 1) 1 ( 1) 1 4 Copyrght 010 Pearso Educato, Ic. Pulshg as Addso-Wesley.

! 6 1 Secto 5. Sgma Notato ad Lmts of Fte Sums 63 7. (a) c 1 & 1 4 8 16 3 () 1 4 8 16 3! 5 0! 4! & (c) 1 1 4 8 16 3! All of them represet 1 4 8 16 3! 6 1 8. (a) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 4 8 16 3 1! 5 & () ( 1) ( 1) Ð ) ( 1) ( 1) ( 1) ( 1) 1 4 8 16 3 0! 3!! & &!! 1 (c) ( 1) Ð ) ( ) ( ) ( 1) ( ) ( 1) 1 4 8 16 3; (a) ad () represet 1 4 8 16 3; (c) s ot equvalet to the other two! 4 ( ) ( 1) ( ) ( ) 1 1 3 1 4 1 3 c c c c 9. (a) 1!! ( ) ( 1) ( ) ( ) 1 0 1 1 1 1 3 0 () 1! 1 c! (c) 1 c ( ) ( 1) ( ) ( ) 1 0 1 3 (a) ad (c) are equvalet; () s ot equvalet to the other two.! 4 1 10. (a) ( 1) (1 1) ( 1) (3 1) (4 1) 0 1 4 9! 3 () ( 1) ( 1 1) (0 1) (1 1) ( 1) (3 1) 0 1 4 9 16 1! (c) ( 3) ( ) ( 1) 9 4 1 3 (a) ad (c) are equvalet to each other; () s ot equvalet to the other two. 6 4 4 1 1 1 11.! 1.! 13.! 5 5 5 1 5 1 1 1 14.! 15.! ( 1) 16.!( 1) 17. (a)! 3a 3! a 3( 5) 15 1 1!! 6 6 6 1 1 () (6) 1 (c)! (a )! a! 5 6 1 1 1 1 (d)!(a )! a! 5 6 11 1 1 1 (e)!( a )!! a 6 ( 5) 16 1 1 1 Copyrght 010 Pearso Educato, Ic. Pulshg as Addso-Wesley.

64 Chapter 5 Itegrato 18. (a)! 8a 8! a 8(0) 0 ()! 50 50! 50(1) 50 1 1 1 1 (c)!(a 1)! a! 1 0 (d)!( 1)!! 1 1 1 1 1 1 1 10 10 19. (a)! 10(10 1) 55 ()! 10(10 1)((10) 1) 385 1 1 10! 10(10 1) 1 (c) 55 305 6 13 13 0. (a)! 13(13 1) 91 ()! 13(13 1)((13) 1) 819 1 1 13! 13(13 1) 1 (c) 91 881 6 7 7 5 1 1 5 1 15 15 15 1 1 1 1 1 1.!! 7(7 ) Š 56.!! 5(5 1) Š 6 6 6 3.! 3! 3! 6(6 )((6) 1) a 3(6) 73 1 1 1 6 6 6 6 4.! 5!! 6(6 )((6) 1) a 5 5(6) 61 1 1 1 6 5 5 5 5 5.! (3 5)! a3 5 3! 5! 5(5 1)((5) 1) 3 Š 5(5 1) 5 Š 40 1 1 1 1 6 7 7 7 7 6.! ( 1)! a!! 7(7 1)((7) 1) Š 7(7 1) 308 1 1 1 1 5 5 5 5 7.!!! Œ Œ! 5(5 1) 5(5 1) Š Š 3376 5 5 5 1 1 1 1 7 7 7 7 6 8. Œ!! Œ!! 7(7 1) 7(7 1) Š Š 588 4 4 4 1 1 1 1 7 500 9. (a)! 3 3a 7 1 ()! 7 7a500 3500 1 1 64 6 (c) Let j Ê j ; f 3 Ê j 1 ad f 64 Ê j 6 Ê! 10! 10 10a6 60 3 j1 36 8 8 8 30. (a) Let j 8 Ê j 8; f 9 Ê j 1 ad f 36 Ê j 8 Ê!! aj 8! j! 8 8a8 1 ) a8 630 9 j1 j1 j1 17 15 () Let j Ê j ; f 3 Ê j 1 ad f 17 Ê j 15 Ê!!aj 15 15 15 15 3 j1! j 4j 4! j! 4j! 15a15 1aa15 1 15a15 1 a 4 4 4a15 j1 j1 j1 j1 140 480 60 1780 6 Copyrght 010 Pearso Educato, Ic. Pulshg as Addso-Wesley.

Secto 5. Sgma Notato ad Lmts of Fte Sums 65 71 (c) Let j 17 Ê j 17; f 18 Ê j 1 ad f 71 Ê j 54 Ê! a 1 54 54 54 54 54! j 17 j 17 1! j 33j 7! a aa a j! 33j! 7 j1 j1 j1 j1 j1 54a54 1aa54 1 54a54 1 6 33 7a54 53955 49005 14688 117648 31. (a)! 4 4 ()! c c 1 1! a!! a 1 1 1 1 (c) 1 1 3. (a)! ˆ 1 ˆ 1 1 ()! c c c (c)! 1 1 1 1 a 1 1 33. (a) () (c) 3 34. (a) () (c) 35. (a) () (c) Copyrght 010 Pearso Educato, Ic. Pulshg as Addso-Wesley.

66 Chapter 5 Itegrato 36. (a) () (c) 37. x x! 1. 0 1., x x 1.5 1. 0.3, x x.3 1.5 0.8, x x.6.3 0.3, ad x x 3.6 0.4; the largest s lpl 1.. & 38. x x! 1.6 ( ) 0.4, x x 0.5 ( 1.6) 1.1, x x 0 ( 0.5) 0.5, x x 0.8 0 0.8, ad x x 1 0.8 0.; the largest s lpl 1.1. & 39. fax x! Let x ad c x. The rght-had sum s! a c! Š ˆ! a 1 1 1 a a! 1. Thus, lm!a c Ä_ 1 lm Œ Ä_ 40. fax x! Let x ad c x. The rght-had sum s! ˆ! )! ) a * * c lm! * * * lm lm Ä_ ˆ Ä_ * Ä_ * Thus,. 41. fax x!! c! ˆ! * (! ( a a Š Let x ad c x. The rght-had sum s a Š Š ( * * a ) ( * ) lm lm Ä_ Ä_ Þ Thus,!ac Œ *. Copyrght 010 Pearso Educato, Ic. Pulshg as Addso-Wesley.

Secto 5. Sgma Notato ad Lmts of Fte Sums 67 4. fax x! Let x ad c x. The rght-had sum s!!! a a c. Thus, lm! c ˆ Ä_ ˆ ˆ ˆ Š lm Œ Ä_. 43. fax x x xa x!! a! Š ˆ!! c c Let x ad c x. The rght-had sum s a a a. Thus, lm!ac c Ä_ Š Š lm Š Œ Ä_ &. 44. fax x x! Let x ad c x. The rght-had sum s! a! Š ˆ!! c c a a a. Thus, lm!a c c Ä_ Š Š lm Š Œ Ä_. 45. fax x 3! Let x ad c x. The rght-had sum s! 3! ˆ 3! 3 a c 4 4 a Š Š a 44. 3 lm!a lm Ä_ Ä_ Thus, c. Copyrght 010 Pearso Educato, Ic. Pulshg as Addso-Wesley.

68 Chapter 5 Itegrato 46. fax x x 3! a Let x ad c x. The rght-! a! Š ˆ ˆ 3 1 1 3 3! 5 Š! 5 3 Š 3 4 1 1! 5!! 1! 3 3 4 5 a a a 1 a a Š 3Š 4Š had sum s c c 5 5 4 6 5 4 1 5 6 3 4 3 4 3 Ä_. Thus, lm!ac c lm. Ä_ 5 6 5 4 1 5 4 1 7 3 4 3 4 1 5.3 THE DEFINITE INTEGRAL! & 0 ( 1. x dx. x dx 3. ax 3x dx 4. dx 5. dx 6. È 4 x x 1 x dx 0! 7. (sec x) dx 8. (ta x) dx c1î 9. (a) g(x) dx 0 () g(x) dx g(x) dx 8 & & & 1Î 0 & (c) 3f(x) dx 3 f(x) dx 3( 4) 1 (d) f(x) dx f(x) dx f(x) dx 6 ( 4) 10 & & & (e) [f(x) g(x)] dx f(x) dx g(x) dx 6 8 & & & (f) [4f(x) g(x)] dx 4 f(x) dx g(x) dx 4(6) 8 16 10. (a) f(x) dx f(x) dx ( 1) * * * * * () [f(x) h(x)] dx f(x) dx h(x) dx 5 4 9 ( ( ( * * * (c) [f(x) 3h(x)] dx f(x) dx 3 h(x) dx (5) 3(4) ( ( ( * (d) f(x) dx f(x) dx ( 1) 1 * ( * * (e) f(x) dx f(x) dx f(x) dx 1 5 6 ( ( * * * (f) [h(x) f(x)] dx [f(x) h(x)] dx f(x) dx h(x) dx 5 4 1 * ( ( ( 11. (a) f(u) du f(x) dx 5 () È3 f(z) dz È3 f(z) dz 5È3 (c) f(t) dt f(t) dt 5 (d) [ f(x)] dx f(x) dx 5 Copyrght 010 Pearso Educato, Ic. Pulshg as Addso-Wesley.

!!! 1. (a) g(t) dt g(t) dt È () g(u) du g(t) dt È Secto 5.3 The Defte Itegral 69!!!!! È g(r) È È È (c) [ g(x)] dx g(x) dx (d) dr g(t) dt Š Š È 1 13. (a) f(z) dz f(z) dz f(z) dz 7 3 4!! () f(t) dt f(t) dt 4 14. (a) h(r) dr h(r) dr h(r) dr 6 0 6 () h(u) du Œ h(u) du h(u) du 6 15. The area of the trapezod s A (B )h (5 )(6) 1 Ê ˆ x 3 dx 1 square uts 16. The area of the trapezod s A (B )h Î Î (3 1)(1) Ê ( x 4) dx square uts 17. The area of the semcrcle s A 1r 1(3) 9 9 1 Ê È9 x dx 1 square uts Copyrght 010 Pearso Educato, Ic. Pulshg as Addso-Wesley.

70 Chapter 5 Itegrato 4 4 18. The graph of the quarter crcle s A 1r 1(4)! 41 Ê È16 x dx 41 square uts 19. The area of the tragle o the left s A h ()(). The area of the tragle o the rght s A h (1)(1). The, the total area s.5 Ê x dx.5 square uts 0. The area of the tragle s A h ()(1) 1 Ê a1 x dx 1 square ut 1. The area of the tragular pea s A h ()(1) 1. The area of the rectagular ase s S jw ()(1). The the total area s 3 Ê a x dx 3 square uts. y 1 È1 x Ê y 1 È1 x Ê (y 1) 1 x Ê x (y 1) 1, a crcle wth È ceter (!ß ) ad radus of 1 Ê y 1 1 x s the upper semcrcle. The area of ths semcrcle s 1 A 1r 1(1). The area of the rectagular ase s A jw ()(1). The the total area s 1 Ê Š 1 1 x dx square uts È 1 Copyrght 010 Pearso Educato, Ic. Pulshg as Addso-Wesley.

x! 4! 3. dx ()( ) 4. 4x dx (4) Secto 5.3 The Defte Itegral 71 5. s ds () a(a) a 6. 3t dt (3) a(3a) a a a 3 a 7. (a) È4 x dx () 4 x 1a 1 È dx 1a 1 0 8. (a) Š 3x È 1 x dx 3x dx È 1 x dx a1a3 a1 1 1 0 0 0 3 1 1 1 4 4 0 0 1 1 Š È È 1 1 1 0 1 a a a a 1a () 3x 1 x dx 3x dx 3x dx 1 x dx 1 3 1 3 1 È Š È Þ& (1) (.5) (0.5)!Þ& 9. x dx 30. x dx 3 1 & È (1) Š 5È Š È 1 31 1 È 31. ) d ) 3. r dr 4 È3 7 Š È3 7!Þ 7 (0.3) 0 3 3! 3 4 33. x dx 34. s ds 0.009 Î ˆ 1Î ˆ 1 1! 3 4! 3 4 35. t dt 36. ) d) a È a (a) Š È3a a 3a a a a 37. x dx 38. x dx a È Š È (3)! 3 3! 3 39. x dx 40. x dx 9 Copyrght 010 Pearso Educato, Ic. Pulshg as Addso-Wesley.

7 Chapter 5 Itegrato 41. 7 dx 7(1 3) 14 4. 5x dx 5 x dx 5 10!! 43. (t 3) dt t dt 3 dt 3( 0) 4 6!! 0 È È È Š È 0 0 44. Š t È dt t dt È dt È È 0 1 1!!! z z 1 3 7 4 45. 1 dz 1 dz dz 1 dz z dz 1[1 ] ˆ ˆ!!!! 46. (z 3) dz z dz 3 dz z dz 3 dz 3[0 3] 9 9 0! 3 0 0 0 1 47. 3u du 3 u du 3 u du u du 3 Š 3 3 ˆ 7 7!! 3 3 3 3 3 3 3 Î Î Î!! 8 48. 4u du 4 u du 4 u du u du 4 3 3 4 3 7 1 ˆ ˆ 7 49. a3x x 5 dx 3 x dx x dx 5 dx 3 5[ 0] (8 ) 10 0!!!! 0 0 3 3! 50. a3x x 5 dx a3x x 5 dx 3 x dx x dx 5 dx!!!! 1 0 1 0 3 7 3 3 3 Š Š 5(1 0) ˆ 5 0 51. Let? x ad let x 0, x? x,! x? xßá ßx c ( 1)? x, x? x. Let the c s e the rght ed-pots of the sutervals Ê c x, c x, ad so o. The rectagles defed have areas: f(c )? x f(? x)? x 3(? x)? x 3(? x) f(c )? x f(? x)? x 3(? x)? x 3() (? x) f(c )? x f(3? x)? x 3(3? x)? x 3(3) (? x) ã f(c )? x f(? x)? x 3(? x)? x 3() (? x) The S! f(c )? x! 3 (? x) 1 1 3(? x)! ( 1)( 1) 3 Š Š 6 1 ˆ 3 ˆ 3! Ä_ Ê 3x dx lm. Copyrght 010 Pearso Educato, Ic. Pulshg as Addso-Wesley.

Secto 5.3 The Defte Itegral 73 0 5. Let? x ad let x 0, x? x,! x? xßá ßx c ( 1)? x, x? x. Let the c s e the rght ed-pots of the sutervals Ê c x, c x, ad so o. The rectagles defed have areas: f(c )? x f(? x)? x 1 (? x)? x 1 (? x) f(c )? x f(? x)? x 1 (? x)? x 1 () (? x) f(c )? x f(3? x)? x 1 (3? x)? x 1 (3) (? x) ã f(c )? x f(? x)? x 1 (? x)? x 1 () (? x) The S! f(c )? x! 1 (? x) 1 1 1? ( x)! ( 1Š Š 1)( 1) 6 1 1 ˆ 3 1 3 1 6 1 ˆ 6! Ä_ 3 Ê x dx lm. 0 53. Let? x ad let x 0, x? x,! x? xßá ßx c ( 1)? x, x? x. Let the c s e the rght ed-pots of the sutervals Ê c x, c x, ad so o. The rectagles defed have areas: f(c )? x f(? x)? x (? x)(? x) (? x) f(c )? x f(? x)? x (? x)(? x) ()(? x) f(c )? x f(3? x)? x (3? x)(? x) (3)(? x) ã f(c )? x f(? x)? x (? x)(? x) ()(? x) The S! f(c )? x! (? x) 1 1?! ( Š Š 1) 1! Ä_ ( x) 1 Ê x dx lm 1. 0 54. Let? x ad let x 0, x? x,! x? xßá ßx c ( 1)? x, x? x. Let the c s e the rght ed-pots of the sutervals Ê c x, c x, ad so o. The rectagles defed have areas: f(c )? x f(? x)? x ˆ?x 1 (? x) (? x)? x f(c )? x f(? x)? x ˆ? x 1 (? x) ()(? x)? x f(c )? x f(3? x)? x ˆ 3? x 1 (? x) (3)(? x)? x ã x f(c )? x f(? x)? x ˆ? 1 (? x) ()(? x)? x!!!! ( 1)????? Š Š 1 1 1 1 ˆ 1 1 ˆ x 1 4 dx lm ˆ ˆ 1.! Ä_ 4 4 The S f(c ) x ˆ ( x) x ( x) x 1 ˆ () Ê Copyrght 010 Pearso Educato, Ic. Pulshg as Addso-Wesley.

74 Chapter 5 Itegrato È È3 0! È È È 3! È3! 55. av(f) Š ax 1 dx Š È3 x dx 1 dx È 3 È Š È3 0 1 1 0. 3 3 56. av(f) ˆ x Š dx ˆ x dx 3 0! 3! 3 3 x 3 6 3 Š ;. 57. av(f) ˆ a 3x 1 dx 1 0! 3 x dx 1 dx 3 Š (1 0).!! 1 3 58. av(f) ˆ a3x 3 dx! 1 0 3 x dx 3 dx 3 Š 3(1 0).!! 1 3 59. av(f) ˆ (t 1) dt! 3! 3! 3! 3 0 t dt t dt 1 dt 3 3 0 3 3 3 3 Š Š (3 0) 1. Copyrght 010 Pearso Educato, Ic. Pulshg as Addso-Wesley.

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