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CHAPTER Itegratio Sectio. Atierivatives a Iefiite Itegratio......... 77 Sectio. Area............................. 8 Sectio. Riema Sums a Defiite Itegrals........... 88 Sectio. The Fuametal Theorem of Calculus.......... 9 Sectio. Itegratio b Substitutio................. 97 Sectio. Numerical Itegratio................... Review Eercises............................. 9 Problem Solvig..............................

CHAPTER Itegratio Sectio. Solutios to O-Numbere Eercises Atierivatives a Iefiite Itegratio. C C 9 9. C. t t 7. t C C t t C t C Give Rewrite Itegrate Simplif 9. C C. C C. C C. C 7. C C C 9. C. C C C. C C. C C C C 77

78 Chapter Itegratio 7. C C C 9.. C C 7 7 C 7 7 C. C. C si cos cos si C cos si C si cos 7. csc t cot t t t csc t C 9. sec si ta cos C t csc t C csc t cot t t ta cos C sec si. ta sec ta C. ta C sec ta f cos C C C. f 7. f C f f ) ) ) f ) f 9. f C f ) ) f ) ),, C C C Aswers will var. f Aswers will var.

Sectio. Atierivatives a Iefiite Itegratio 79.. cos,, cos si C si C C si (a) Aswers will var.,, C 8 C C. f, f 7. f C f C C f ht 8t, h ht 8t t t t C h C C ht t t 9. f. f f f f f f C f C C f f C f C C f f C C f C C f f C f C C f. (a) ht.t t.7t t C h C C ht.7t t h.7 9 cm

8 Chapter Itegratio. f. Graph of is give. f (a) f. No. The slopes of the taget lies are greater tha o,. Therefore, f must icrease more tha uits o,. (c) No, f < f because f is ecreasig o,. () f is a maimum at. because f. a the first erivative test. (e) f is cocave upwar whe is icreasig o, a,. f is cocave owwar o,. Poits of iflectio at,. f (f) (g) f is a miimum at. 8 7. at ftsec vt t t C v C st t t t t C s C st t t Positio fuctio The ball reaches its maimim height whe vt t t t 8 secos s 8 8 8. feet 9. From Eercise 8, we have: st t v t st t v whe maimum height. v s v v v v v v, v 87.7 ftsec t v time to reach 7. 7. at 9.8 7. From Eercise 7, ft.9t t. v t 9.8t (Maimum height whe v. ) vt 9.8 t 9.8t C 9.8t v v C vt 9.8t v t f t 9.8t v t.9t 9.8 v t C f s C f t.9t v t s a. f 7. m 9.8 vt. t.t v.t, sice the stoe was roppe, v. st.t t.8t s s.8 s s Thus, the height of the cliff is meters. vt.t v msec

Sectio. Atierivatives a Iefiite Itegratio 8 77. t t t 9t t 79. vt t t > t (a) vt t t t 9 t t t t t vt t t C at vt t t C C vt > whe < t < or < t <. t t positio fuctio (c) at t whe t. at vt v t acceleratio t 8. (a) v kmhr v 8 kmhr 8 8 msec at a costat acceleratio vt at C v v 8 st a t a s 7 vt at a a 7.7 msec 8 t s 89.8 m msec 8. Truck: Automobile: At the poit where the automobile overtakes the truck: (a) vt st t Let s. t t at t t vt t Let v. st t Let s. tt whe t sec. s ft v ftsec mph 8. mihr8 ftmi sechr ftsec (a) t V ftsec.7.7.7. 9. V t.8t.t.79 V t.8t.799t.77 V ftsec.8.7 7.8 88 9.87 9. (c) S t V t t.8 t. t.79t S t V t t.8t I both cases, the costat of itegratio is because S S S 9. feet S 97. feet.799t.77t The seco car was goig faster tha the first util the e.

8 Chapter Itegratio 87. at k vt kt st k sice v s. t At the time of lift-off, kt a kt.7. Sice kt.7, t. k v. k k. k.k k. 8,8.7 mihr 7. ftsec. 89. True 9. True 9. False. For eample, because C C C 9. f,, f C, C, f C C f is cotiuous: Values must agree at : C C, f, < < < The left a right ha erivatives at o ot agree. Hece f is ot ifferetiable at. Sectio. Area. i. k 7. 9 i i i i i k 7 i 9. 8 8 8 j j 8.. i i 7. i i. k c c c c c c i i i. i i 9 i i 99 7 i i

Sectio. Area 8 9. i ii i,,8, i i i i. sum seq,,,, 9 (TI-8) i > i i 9. S 9.. s 9. S s 7 7. 9. S.78 8 s.8 8 S 7 8 9 7 8.7 9 s 7 8 9 7 8 9.. lim 8 8 8 8 lim. lim 8 8 9 8 lim. i i i i S S. S. S. S,. 7. kk k k k k S.98 S.9998 S.999998 S,.99999998 S 9. lim 8 lim 8 lim lim i i lim i i 8

8 Chapter Itegratio. lim i i lim lim i i lim lim. lim lim lim lim i i i i i. (a) (e) s..8.9.98 S.... Epoits: < < <... < < (c) Sice is icreasig, f m i f i o i, i. s () f M i f i o i, i S f i i i i f i f i i i i f i i i (f) lim i i lim lim lim i i lim i lim i i lim lim i 7. o,. Note: s i f i i i i i Area lim s 9. o,. Note: S f i i i i i i Area lim S 7

Sectio. Area 8. o,. Note: s i f i i i i i i 8 8 8 Area lim s 8 7. o s i i,. Note: f i i i 7i 7i 7 9i 7 9 7 89 8 8 7 8 7 Area lim s 89 7 8.. o,. Note: Agai, T is either a upper or a lower sum. T i i i f i i i i i i i 8i i i i 8i i i i i i i i Area lim T

8 Chapter Itegratio 7. f, Note: S f m i i i i i f i i i Area lim S lim 9.. f, S i f i i i 7 Note: 7 7 Area lim S lim 9 9 9 S i i i i i i i g i i Area lim S 8 i i 9 7 9 9 g,. Note: i i i 8i i 8i 8 8 8. f,,. Let c i i i. c c c c 7,,,, Area f c i c i i i 9 9 9 8 7. f o,. 8 f ta,, Let c i i i. Area f c i ta c i i i c c c,,,, ta ta c 7 7 ta ta. Approimate area.88..9..8 Eact value is

Sectio. Area 87 9. f ta o,. 8 8 Approimate area..87.8.. 7. We ca use the lie boue b a a b. The sum of the areas of these iscribe rectagles is the lower sum. The sum of the areas of these circumscribe rectagles is the upper sum. a b a b We ca see that the rectagles o ot cotai all of the area i the first graph a the rectagles i the seco graph cover more tha the area of the regio. The eact value of the area lies betwee these two sums. 7. (a) 8 8 (c) Lower sum: s. 8 Upper sum: S.7 () I each case,. The lower sum uses left epoits, i. The upper sum uses right epoits, i. The Mipoit Rule uses mipoits, i. (e) Mipoit Rule: M 7 9 9. 8 s S M. 7.8 8.9 8.99 9..7.8 9.79 9. 9.88 9. 9. 9.7 9. 9. (f) s icreases because the lower sum approaches the eact value as icreases. S ecreases because the upper sum approaches the eact value as icreases. Because of the shape of the graph, the lower sum is alwas smaller tha the eact value, whereas the upper sum is alwas larger.

88 Chapter Differetiatio 7. 77. True. (Theorem. ()) b. A square uits 79. f si,, Let A area boue b f si, the -ais, a. Let A area of the rectagle boue b,,, a. Thus, A.779. I this program, the computer is geeratig N pairs of raom poits i the rectagle whose area is represete b A. It is keepig track of how ma of these poits, N, lie i the regio whose area is represete b A. Sice the poits are raoml geerate, we assume that A A N N A N N A...7.. f ( ) = si( ) π π ( ) π, The larger N is the better the approimatio to A. 8. Suppose there are rows i the figure. The stars o the left total..., as o the stars o the right. There are stars i total, hece....... 8. (a).9..7.9 (c) Usig the itegratio capabilit of a graphig utilit, ou obtai A 7,897. ft. Sectio. Riema Sums a Defiite Itegrals. f,,,, c i i i i i i = lim fc i i lim i i i i lim lim i i i () )...... ( lim.

Sectio. Riema Sums a Defiite Itegrals 89. o,. f c i i f i i i. o,. Note:, as lim i i i i i i i i Note:, as f c i i f i i i i i i i 8i i lim 7. o,. Note:, as 9. lim f c i i f i i i i i i i i i i i lim i c i i o the iterval,. i. lim i c i i o the iterval,... 7. 9. si.. Rectagle A bh A Rectagle

9 Chapter Itegratio. Triagle A bh A 8 Triagle 7. Trapezoi A b b h 9 A 9 Trapezoi 9. Triagle A bh Triagle A. Semicircle A r A 9 9 Semicircle I Eercises 9,,. 7. 8, 8 8 9... (a) 7 (c) f () f f 7 f f f f f. (a) (c) g g () f f f g f g 8 g f g f. (a) Quarter circle below -ais: r Triagle: bh (c) Triagle Semicircle below -ais: () Sum of parts a (c): (e) Sum of absolute values of a (c): (f) Aswer to () plus : 7. The left epoit approimatio will be greater tha the actual area: > 9. Because the curve is cocave upwar, the mipoit approimatio will be less tha the actual area: <

Sectio. Riema Sums a Defiite Itegrals 9. f. is ot itegrable o the iterval, a f has a iscotiuit at. a. A square uits.. si 7. L M R 9. si 8.8.99.77..77.8.7.88.7.9.8.99.77..77 L M R 8.89.87.799.7.7.78.78.78.78.78.987.88.88.8.87. True. True. False 7. f,, 8,,, 7, 8,,, c, c, c, c 8 f c i f f f f 8 i 88 7

9 Chapter Differetiatio 9. f,, is ratioal is irratioal is ot itegrable o the iterval,. As, f c i or f c i i each subiterval sice there are a ifiite umber of both ratioal a irratioal umbers i a iterval, o matter how small. 7. Let f,, a i. The appropriate Riema Sum is f c i i i i i i lim... lim i. lim lim Sectio. The Fuametal Theorem of Calculus. f is positive.. f. 7. 9. t t t t. t t t t t t t t.. u u u u u u u u 7. t t t t 9. 8. t t t t t 7. split up the itegral at the zero 9 9 9 9 9 9 9 9 9

Sectio. The Fuametal Theorem of Calculus 9. 8 8 9 8 8 7. 9... 7. A si cos sec ta sec ta sec,t t, t t $,. A 9. A cos si. Sice o,,. Sice o,,. A. A 8. 8 f c c c c c c 8 c ± c ± c.8 or c.798

( ( 9 Chapter Itegratio 7. sec ta f c sec c 8 sec c sec c ± c ±arcsec ±arccos ±.87 9. 8 8 8 8 8 8 (, ( 8, Average value 8. whe 8 8 si cos Average value si.9,. or ±. f area of regio A. 9. f f. (a) F k sec. F k F sec.. ±.. (.9, π (., π ( (. If f is cotiuous o a, b a F f o a, b, b the f Fb Fa. a 7. f f f... sec.79t.t.7t t.8t.7t.9t.8 liter ta 8.99 ewtos 87 ewtos

Sectio. The Fuametal Theorem of Calculus 9. (a) The area above the -ais equals the area below the -ais. Thus, the average value is zero. The average value of S appears to be g. 7. (a) v 8. t.78t.8t.9 9 9. 7. (c) 7 vt t 8. t.78t.8t.9t F t t t t F 8 77. (a) F F8 8 8 F cos si F si si.78 F si si.8 F8 si 8 si.79 7. si si t t 8 t 8 7. (a) 7 meters F v v F F 8 F8 7 8 79. (a) v v v t t t t sec t t ta t ta ta sec 8. F t t t 8. F t t 8. F t cos t t F F F cos

9 Chapter Itegratio 87. 89. 9. F t t F 8 t t 8 Alterate solutio si F t t si F t t si t si 9. F si cos cos si F si si si cos g f t t g, g, g, g, g f g Alterate solutio: F t t t t t t t t t t F 8 9. (a) F si t t F si si C t t t C $7, C $,7 C $8,9 g has a relative maimum at. 97. True 99. False;. f t t t t B the Seco Fuametal Theorem of Calculus, we have f. Sice f, f must be costat. Each of these itegrals is ifiite. f has a oremovable iscotiuit at.

Sectio. Itegratio b Substitutio 97.. t t t 9t t t t 9 t t t t Total istace tt t t t t t t t t t 8 uits Total istace tt vt t t t t t t t uits Sectio. Itegratio b Substitutio f gg u g u g... ta sec ta sec 7. 9. 9 C C 9 C 9 C 9 C 9 9

98 Chapter Itegratio. C C C. C C C. tt t t t t t t t C t t C t t t C 7. C 8 C 8 C 8 9. C C C. C C C. C C C. t t t t t t t C t t C t t t t 7. C C C

Sectio. Itegratio b Substitutio 99 9. 7 7 C C C 7. t t t t t t t t t C t t t C t t t t t. 9 9 9 C C C C 9 9. 7. C C C C 9. (a),, C C, : C C. si cos C. si si cos C. cos cos si C

Chapter Itegratio 7. si cos cos OR si si C si C si cos cos si cos C cos C OR si cos si cos si cos C 8 9. ta sec ta C ta C. csc cot cot csc cot C cot C ta C sec C sec C. cot csc cot C. f cos si C Sice f si C, C. Thus, f si. 7. u, u, u u u u u u u u u C u u C C C 9. u, u, u u u u u u u u u u 7 u7 C u u u C C 8 C

Sectio. Itegratio b Substitutio. u, u, u u u u 8u u u u 8u u u u 8 u u u C u u u C C 8 C C. u, u, u u u u u uu u u u u u u C u u C C C C where C C. u. Let u, u. 8 7. Let u, u 9 9 7 8 9

Chapter Itegratio 9. Let u, u. 9 7. Let u, 9 u. 9 9 7. u, u, u Whe, u. Whe, u. u u u 7. cos si u u u u u 77. u, u, u Whe, u. Whe 7, u 8. 7 8 Area u u u 79. A 8. Area 8. 8 u u u 7 u7 u 8 si si cos cos sec. sec ta 8. 8 7 7 9 8 7 8.8 87. cos 7.77 8 89. C C C The iffer b a costat: C C.

Sectio. Itegratio b Substitutio 9. f is eve. 9. f is o. (c) 8 7 9. the fuctio is a eve fuctio. 8 ; (a) 8 8 () 97. 99. Aswers will var. See Guielies for Makig a Chage of Variables o page 9. 8. f is o. Hece,.. V t k t Vt k t t k t C V k C, V k C, Solvig this sstem iels k, a C,. Thus, Vt, t,. Whe t, V $,.. b b a 7..7 si (a) (c) a 7.t. 7.t. 7.t. t t b a t cos.. t cos. 7 7.t. t cos. 89. thousa uits.. thousa uits. cos t b a 7. thousa uits

Chapter Itegratio 7. b a (a) (c) a b sit cost t b a cost cost cost sit sit sit cost b sit a.8 amps amps.7 amps 9. False C. True. True a b c a c O si cos si cos C b Eve b. Let u h, the u. Whe a, u a h. Whe b, u b h. Thus, bh bh f h f u u f. b a ah ah Sectio.. Eact: Trapezoial: Simpso s:. Eact: Trapezoial: Simpso s: Numerical Itegratio 8.7 8. 7.7.7..

Sectio. Numerical Itegratio. Eact: Trapezoial: Simpso s: 7. Eact: Trapezoial: Simpso s: 9. Eact: Trapezoial: Simpso s:. 9 8 7 8.. 9 7 8. 9 9 8 8.7 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 8 7 9 8.7 8 8 9.7 7 9 8 8 9.7.7. Trapezoial: Simpso s: Graphig utilit:. 8 78.8 8 78.. Trapezoial: Simpso s: 8..7 Graphig utilit:.9

Chapter Itegratio. Trapezoial: Simpso s: Graphig utilit:.977 cos 8 cos cos cos cos.97 cos cos cos cos cos.978 cos cos. 7. Trapezoial: si 8 si si. si. si.7 si..89. Simpso s: si si si. si. si.7 si..89 Graphig utilit:.89 9. Trapezoial: ta ta ta ta Simpso s: ta 8 ta ta ta Graphig utilit:.8.9.8. (a). f a b = f() The Trapezoial Rule overestimates the area if the graph of the itegra is cocave up. f f f f (a) Trapezoial: Error. sice f is maimum i, whe. Simpso s: Error f. sice 8. f i,. f (a) is maimum whe a Trapezoial: Error f i, f f. is maimum whe a whe <., >,., >.; let. f. 8 <., Simpso s: Error >,.7, >.; let.

Sectio. Numerical Itegratio 7 7. f (a) f i,. f is maimum whe a 8 Trapezoial: Error <., >,.7, > 9.; let. f f i, 7 is maimum whe a f. f. Simpso s: Error <., >,.7, >.; let. 8 9. f ta (a) f sec ta i,. is maimum whe a f 9.. f Trapezoial: Error 9. <., >,7.7, >.; let. f 8 sec ta ta 8 ta i, f is maimum whe a f 98.7. Simpso s: Error <., >,,8., > 7.; let 8. 8 98.7. Let f A B C D. The f. b a Simpso s: Error 8 Therefore, Simpso s Rule is eact whe approimatig the itegral of a cubic polomial. Eample: This is the eact value of the itegral.. f o,. L M R T S.777.9 8...8 8.88.8.9...9..97.88.8.8.78..8.7.77..8.7.7.9.8.7.7.7

8 Chapter Itegratio. f si o,. L M R T S.8..7.79.99 8.89...8..78.99..9..99.9.9..7..9.7.8.7.7.888...79 7. A cos Simpso s Rule: cos 8 cos 8 cos 8 cos cos 8 8... cos.7 π π 9. W Simpso s Rule:...,.8 ft lb. Simpso s Rule,.9.8.98...98.98.. Area 9 9 9 88 7 89, sq m. t si, B trial a error, we obtai t.77.

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