CHAPTER 5 INTEGRATION

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1 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.

2 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.

3 Secto 5.1 Area ad Estmatg wth Fte Sums 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) mles () Roughly, after hours, the car would have goe mles, where hours.7 sec. At.7 sec, the velocty was approxmately 10 m/hr. Copyrght 010 Pearso Educato, Ic. Pulshg as Addso-Wesley.

4 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 [ ](1) ft/sec () Usg rght ed-pots we ota a lower estmate: speed [ ](1) 45.8 ft/sec (c) Upper estmates for the speed at each secod are: t v Thus, the dstace falle whe t 3 secods s s [ ](1) 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 v 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 [ ](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 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) ˆ ˆ ˆ 3 ˆ ˆ 5 ˆ ˆ 7 ˆ 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 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 The heghts of the four approxmatg rectagles are 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 ( ) average value Partto [0ß4] to the four sutervals [0ß1], [1ß ß], [ß3], ad [3ß4]. The mdpots of the sutervals 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 (to 5 decmal places), f(m ) 1 Š cos Š 1 ˆ cos ˆ , f(m ) 1 Š cos Š 1 ˆ cos ˆ 51 1 ˆ 3 1 ˆ , ad f(m ) 1 Š cos Š 1 ˆ cos ˆ The wdth of each rectagle s 1 ˆ ? x. Thus, Area (0.7145)(1) ( )(1) ( )(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.

5 Secto 5.1 Area ad Estmatg wth Fte Sums 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 ( ) 363 gal, lower estmate ( ) 1693 gal. (c) worst case: t 5,000 Ê t 31.4 hrs; est case: t 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 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 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() ! ! cos 1 cos(1 1) cos( 1) cos(3 1) cos(4 1) ! s 1 s(1 1) s( 1) s(3 1) s(4 1) s(5 1) ! È3 È ( 1) s ( 1) s ( 1) s ( ) s 0 1! ( 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.

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

8 64 Chapter 5 Itegrato 18. (a)! 8a 8! a 8(0) 0 ()! 50 50! 50(1) (c)!(a 1)! a! 1 0 (d)!( 1)!! (a)! 10(10 1) 55 ()! 10(10 1)((10) 1) ! 10(10 1) 1 (c) (a)! 13(13 1) 91 ()! 13(13 1)((13) 1) ! 13(13 1) 1 (c) !! 7(7 ) Š 56.!! 5(5 1) Š ! 3! 3! 6(6 )((6) 1) a 3(6) ! 5!! 6(6 )((6) 1) a 5 5(6) ! (3 5)! a3 5 3! 5! 5(5 1)((5) 1) 3 Š 5(5 1) 5 Š ! ( 1)! a!! 7(7 1)((7) 1) Š 7(7 1) !!! Œ Œ! 5(5 1) 5(5 1) Š Š Œ!! Œ!! 7(7 1) 7(7 1) Š Š (a)! 3 3a 7 1 ()! 7 7a (c) Let j Ê j ; f 3 Ê j 1 ad f 64 Ê j 6 Ê! 10! 10 10a j (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 ) a j1 j1 j () Let j Ê j ; f 3 Ê j 1 ad f 17 Ê j 15 Ê!!aj j1! j 4j 4! j! 4j! 15a15 1aa a15 1 a 4 4 4a15 j1 j1 j1 j Copyrght 010 Pearso Educato, Ic. Pulshg as Addso-Wesley.

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

10 66 Chapter 5 Itegrato 36. (a) () (c) 37. x x! , x x , x x , x x , ad x x ; 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 , ad x x ; the largest s lpl 1.1. & 39. fax x! Let x ad c x. The rght-had sum s! a c! Š ˆ! a 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.

11 Secto 5. Sgma Notato ad Lmts of Fte Sums 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 lm!a lm Ä_ Ä_ Thus, c. Copyrght 010 Pearso Educato, Ic. Pulshg as Addso-Wesley.

12 68 Chapter 5 Itegrato 46. fax x x 3! a Let x ad c x. The rght-! a! Š ˆ ˆ ! 5 Š! 5 3 Š ! 5!! 1! a a a 1 a a Š 3Š 4Š had sum s c c Ä_. Thus, lm!ac c lm. Ä_ 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) (a) f(x) dx f(x) dx ( 1) * * * * * () [f(x) h(x)] dx f(x) dx h(x) dx ( ( ( * * * (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 ( ( * * * (f) [h(x) f(x)] dx [f(x) h(x)] dx f(x) dx h(x) dx * ( ( ( 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.

13 !!! 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 (a) f(z) dz f(z) dz f(z) dz 7 3 4!! () f(t) dt f(t) dt (a) h(r) dr h(r) dr h(r) dr () h(u) du Œ h(u) du h(u) du 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) Ê È9 x dx 1 square uts Copyrght 010 Pearso Educato, Ic. Pulshg as Addso-Wesley.

14 70 Chapter 5 Itegrato 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.

15 x! 4! 3. dx ()( ) 4. 4x dx (4) Secto 5.3 The Defte Itegral 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 (a) Š 3x È 1 x dx 3x dx È 1 x dx a1a3 a Š È È a a a a 1a () 3x 1 x dx 3x dx 3x dx 1 x dx È Š È Þ& (1) (.5) (0.5)!Þ& 9. x dx 30. x dx 3 1 & È (1) Š 5È Š È È 31. ) d ) 3. r dr 4 È3 7 Š È3 7!Þ 7 (0.3) 0 3 3! x dx 34. s ds Î ˆ 1Î ˆ 1 1! 3 4! 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! x dx 40. x dx 9 Copyrght 010 Pearso Educato, Ic. Pulshg as Addso-Wesley.

16 7 Chapter 5 Itegrato dx 7(1 3) x dx 5 x dx 5 10!! 43. (t 3) dt t dt 3 dt 3( 0) 4 6!! 0 È È È Š È Š t È dt t dt È dt È È 0 1 1!!! z z 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! u du 3 u du 3 u du u du 3 Š 3 3 ˆ 7 7!! Î Î Î!! u du 4 u du 4 u du u du ˆ ˆ a3x x 5 dx 3 x dx x dx 5 dx 3 5[ 0] (8 ) 10 0!!!! ! 50. a3x x 5 dx a3x x 5 dx 3 x dx x dx 5 dx!!!! Š Š 5(1 0) ˆ 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.

17 Secto 5.3 The Defte Itegral 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) ˆ ˆ 6! Ä_ 3 Ê x dx lm 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 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 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.

18 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 È av(f) ˆ x Š dx ˆ x dx 3 0! 3! 3 3 x Š ;. 57. av(f) ˆ a 3x 1 dx 1 0! 3 x dx 1 dx 3 Š (1 0).!! av(f) ˆ a3x 3 dx! x dx 3 dx 3 Š 3(1 0).!! av(f) ˆ (t 1) dt! 3! 3! 3! 3 0 t dt t dt 1 dt Š Š (3 0) 1. Copyrght 010 Pearso Educato, Ic. Pulshg as Addso-Wesley.

CHAPTER 5 INTEGRATION

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