CHAPTER 4 Integration

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

2 CHAPTER Itegratio Sectio. Solutios to Eve-Numbere Eercises Atierivatives a Iefiite Itegratio.. C. r r C C 8. C C Give Rewrite Itegrate Simplif. C C C C c. C C. 9 9 C 9 C. C 8. C C C. C C. C C C

3 Sectio. Atierivatives a Iefiite Itegratio. 7 7 C. C C C. C t t t t t C C C t t t C t t t t C t t t. tt t t t t t t C. t t C t t t C t t tt t C t.. t si t t t cos t C 8. t t cos t C t si t sec ta sec sec ta sec. sec ta C sec ta C sec ta sec sec ta sec sec ta C cos cos cos ta C sec si csc cot csc C cos csc C csc cot si si cos cos si cos si. f. f 8. f C = f C f C C = C = 8 8 f () = + f ()= f f() = + f f()=

4 Chapter Itegratio.,,.,, C C C C C C, >. (a) (b),, (, ) C C C C 7 7. g, g 8. fs s 8s, f g C fs s 8s s s s C g C C f C C C g fs s s. f. f si f f f f f C f C C f f C f C C f f si cos C f C C f cos f cos si C f C C f si

5 Sectio. Atierivatives a Iefiite Itegratio. P kt, t. Sice is egative o,, is ecreasig o t,. Sice is positive o,, is icreasig Pt kt t o,. has a relative miimum at,. Sice is kt C positive o,, f is icreasig o,. f f f f f f P C C P k k f f Pt t t P7 7 bacteria f 8. ft at ftsec 7. f v f s ft vt t t C f C v C v ft t v f t st t v t t v t C v ftsec s ft (a) t t Choosig the positive value, t 7 st t t t ± 7. secos. f C s C s f t t v t s (b) vt st t v ftsec 7. From Eercise 7, ft.9t. (Usig the 7. From Eercise 7, f t.9t v t. If cao floor as positio.) f t.9t v t, f t.9t the.9t vt 9.8t v t.9 t. 8. sec for this t value. Hece, t v 9.8 a we solve v v v v v v 9.8 v v v 88.8 v. msec.

6 Chapter Itegratio 7. v v GM 78. v GM C Whe R, v v. v GM R C t t t (a) t 7t t 9 at vt t t vt t t t t t (b) vt > whe < t < a < t <. C v GM R v GM v GM R v GM v GM R v v GM R (c) at t whe t 7. v (a) at cos t vt at t cos t t si t C si t sice v f t vt t si t t cos t C f cos C C C f t cos t (b) vt si t for t k, k,,, v mph ftsec mph ftsec mph ftsec at a vt at st a t t Let s. vt after car moves ft. at whe t a. s a a a a whe a at. vt.t st 8.t t.. (a).t (b).t (c) mph = ft/sec t.. s 7. ft. t.7. s 7. ft. mph = ft/sec feet feet mph = ft/sec mph It takes. secos to reuce the spee from mph to mph,. secos to reuce the spee from mph to mph, a. secos to reuce the spee from mph to mph. Each time, less istace is eee to reach the et spee reuctio.

7 8. No, car will be ahea of car. If v a are the respective velocities, the v t t Sectio. Atierivatives a Iefiite Itegratio v t t > v t t. 8. (a) v.9t.t.9t.988 (b) st vtt.9t.t.9t.988t Note: Assume s is iitial positio s 9. feet 88. Let the aircrafts be locate a 7 miles awa from the airport, as iicate i the figure. v A t k A t v B k B t Airport A B s A t k A t t s B k B t t 7 (a) Whe aircraft A las at time s A t A k A t A t A t A ou have v A t A k A t A k A t A t A t A t A t A t A. k A t A S At S t t t Similarl, whe aircraft B las at time ou have v B t B k B t B k B t B s B t B k B t B t B 7 k B t B t B 7 9,7 t B t B t B t B 7 t B. S B t S t 9,7 t 8 t 7 7 (b) (c) s B t s A t Yes, < for t >.. s A s B.. 9. True 9. True

8 Chapter Itegratio 9. False. f has a ifiite umber of atierivatives, each ifferig b a costat. 9. s c ss cc Thus, s c k for some costat k. Sice, s a c, k. Therefore, s c. sc cs [Note that s si a c cos satisf these properties.] Sectio. Area. k kk.. i i i j j 7 8. i i. j j. i i. i i. i i i i 8. 9 i i i i i 7. i ii i i. sum seq,,,,, (TI-8) i i 8 > i i i,. S. s S s S s

9 Sectio. Area 7. S 9.89 s.9. lim lim. lim lim. j j j j S S. S. S. S,. 8. i i i i i i S. S. S.7 S,.7 S. lim lim lim lim i. i i i lim i i lim lim lim i i i i i i i lim

10 8 Chapter Itegratio. lim i i lim lim lim i i i i 8i i lim lim 8. (a) (b) Epoits: < < <... < < < <... < < (c) Sice is icreasig, f m i f i o i, i. () (e) s f M i f i o [ i, i S f i i i i i i f i f i i f i s S i i (f) lim i i lim lim i i lim lim lim lim lim

11 Sectio. Area 9 8. o,. S f i i i i Note: i i Area lim S. o S l 7 f i l,. Note: l l l i Area lim S 9. o,. Fi area of regio over the iterval,. s i f i i i Area lim s i i Note: Area. o,. Note: Sice both icreases a ecreases o,, T is either a upper or lower sum... T i f i i i i i i i i..... Area lim T

12 Chapter Itegratio. o,. Note: s i i i f i i i i i i i i i i i i Area lim s 7 8. g,. Note:. S Area lim S i i g i i i i f,. Note: S i i i f i i i i i i i i i Area lim S. h, Note: S i h i i i i i i i 8 Area lim S 9

13 . f,,. Let c i i i., c, Area i f 8 o,. c c c 7,, f c i c i c i i 8 f si,, Let c i i i. Area i f c i si c i 8 si c 8,, c i si Sectio. Area c,, 8 c 7 7 si si. Approimate area f cos o,. 8 Approimate area See the Defiitio of Area. Page f, 8 s S M (Note: eact aswer is.) True. (Theorem.) a. A square uits

14 Chapter Itegratio 8. (a) (b) si h rθ h r si A bh rr si r si r r h (c) A r si r Let. As,. si r si lim A lim r si r r 8. (a) i i The formula is true for : Assume that the formula is true for k: k i kk. i The we have k i i k i k i kk k k k Which shows that the formula is true for k. (b) i i The formula is true for because Assume that the formula is true for k: k i k k i The we have k i i k i k i k k k k k k k k which shows that the formula is true for k. Sectio. Riema Sums a Defiite Itegrals. f,,,, c i i i i i i i lim f c i i lim i i lim lim lim lim i i i i i i i lim

15 Sectio. Riema Sums a Defiite Itegrals. o,. Note:, as f c i i f i i i i i. o lim,. Note: fc i i f i i i i i i i lim 8 i, as i i 8. o,. Note:. i i i i lim 7 7 lim fc i i f i i i i o the iterval, c i c i i 9i ; as 7.. lim 8. i c i i o the iterval,.. ta

16 Chapter Itegratio.. Rectagle A bh a a A 8a a. Triagle A bh A Rectagle Triagle a a 8. Triagle A bh 88 A 8 8. Triagle A bh aa a A a a a. Semicircle A r r A rr r a Triagle r Semicircle 8 a a r r r 8 I Eercises,,. 8.. (a) (b),. 8. (c) f () f f f f f f f.. (a) (b) (c) f () f f f f f f f f

17 . (a) (c) f f (b) f f 8 f eve Sectio. Riema Sums a Defiite Itegrals () f f f f o Let u. 8. The right epoit approimatio will be less tha the actual area: <. The average of Eercise 9 a Eercise cosists of a trapezoial approimatio, a is greater tha the eact area: >. is itegrable o,, but is ot cotiuous o,. There is iscotiuit at. To see. that f is itegrable, sketch a graph of the regio boue b f a the -ais for. You see that the itegral equals. b. A square uits c. Area 7.. si 8. L M R L M R False. True. False

18 Chapter Itegratio 8. f si,,,,,,,,, c c c,,, i f c i i f f f f c 7. To fi, use a geometric approach..78 Thus,. Sectio. The Fuametal Theorem of Calculus. f cos cos. f is egative v v 7 8. v v v v t 9t t t 9 t 9 9. u u u u u

19 . v v v tt t t t t t t Sectio. The Fuametal Theorem of Calculus 7 8 tt t 8... P si cos csc cot t cos t t t si t si cos. A 8. A si cos.7% 8. A split up the itegral at the zeros,

20 8 Chapter Itegratio. Sice o. 8, 8 Area 8. Sice o,,. A f c 9 c 9 c cos si f c Average value cos c c ± (a) f Sum of the areas cos A cos. si A A A A 8 A A A 7.. (.88, π.7 ( c 9. (b) Average value (c) A 8 Average value f 7 8

21 . f area or regio B f f... Sectio. The Fuametal Theorem of Calculus 9 8. f. Average value f R..8. R. P t (a) t P f.. kr r r k R R r r Average profit (b) t t.. t t (c) The efiite itegral iels a better approimatio.. (a) R.t.7t.7t 7.7t (b) (c) 8.97 R kr Rt t.t.7t.7t 7.7t 8. (a) histogram N t (b) customers (c) Usig a graphig utilit, ou obtai Nt.87t.9t () 9 (e) Nt t 8. The estimate umber of customers is 8.. (f) Betwee P.M. a 7 P.M., the umber of customers is approimatel Hece, 7. per miute. 7 Nt t.8 7.

22 7 Chapter Itegratio 7. t F t t t t t F F 7. Note: F t t t 7. F8 8 8 F t t t t t F F. 7. F8 F si cos F cos. F cos.7 F8 cos (a) tt t (b) 78. (a) 8. (b) cos cos cos t t t t t t t t 8 8. (a) F (b) sec t ta t t sec t sec sec sec ta t F t t 8. F t t 8. F F sec t t F sec

23 t F t t F F t t t t Alterate solutio: F 9. F si F si si Alterate solutio F t t Sectio. The Fuametal Theorem of Calculus 7 t t t t t t t t F 8 F 9. (a) (b) g (c) Miimum of g at,. () Miimum at,. Relative maimum at,. (e) O, g icreases at a rate of (f) Zeros of g:, (a) gt t lim gt t Horizotal asmptote:. (b) A t t t t 8 8 lim A lim 8 8 The graph of A oes ot have a horizotal asmptote. 98. True. Let Ft be a atierivative of f t. The, v v f t t u Ft Fv Fu u v u f t t Fv Fu Fvv Fuu f vv f uu.

24 7 Chapter Itegratio s. G s f t t s s (a) G f t t s (c) G f () G f s f t t f t t s (b) Let Fs s f t t. G Fs s G F f t t G f t t. t t t t 7t t 9 t t t Usig a graphig utilit, Total istace tt 7.7 uits Sectio. Itegratio b Substitutio f gg u g u g.. sec ta cos. si cos si C 9 C 9 9. C C. C C C C C C 8

25 Sectio. Itegratio b Substitutio 7. t t t t t t t C t C t t C t t t t 8. u u u u u u u C u 9 u u C 9 9 u u u u C. C C C. C C C.. C C C 9 C C 9 C C C C t t t C t t t t t t C t t C t t t C t t t t t. t t t t t t t t C t t C t t t C t t

26 7 Chapter Itegratio C 7 C 7 C 7 7 C C C C 8 C. (a) (b) cos,, cos cos si C, : si C C si. si si cos C. cos cos si C. si si cos C 8. sec ta sec ta sec C. ta sec ta C ta C. si cos cos cos si C cos C sec C. csc csc cot C. f sec ta sec C Sice f sec C, C. Thus f sec.

27 Sectio. Itegratio b Substitutio 7 8. u, u, u. u, u, u u u u u u u u u C u u C C C uu u u u u u u C u u C C C C. Let u, u, u.. u u u u t, t u, t u t t t u u u u 7u u u u u u u C 7 u7 u C u u C C u u 7 C 7 7 t t 7 C C 7 t t C. Let u 8, u ,7 8. Let u, u. 7. Let u, u.

28 7 Chapter Itegratio 7. Let u, 7. Let u, u, u. Whe, u. Whe, u 9. 9 u u u 9 u u u u u u cos si u, u, u Whe, u. Whe, u 8. 8 Area u u u 8. A si cos cos si 8 u 7 u u u u 7 u7 u Let u, u. Area csc cot csc cot csc si. 9. si cos si cos si si cos cos si cos cos C si C C si C C The iffer b a costat: C C.

29 Sectio. Itegratio b Substitutio f si cos is eve. 9. f si cos is o. si cos si cos si cos si 9. (a) si sice si is smmetric to the origi. (b) (c) 98. cos cos cos si cos si sice cos is smmetric to the -ais. () si cos sice si cos si cos a hece, is smmetric to the origi. si cos si cos cos si. If u the u a u, u.. Q t k t. Qt k t t k t C R..99 si.t.77 (a) 8 Q C Qt k t. (a) Q k,, k Thus, Qt t. Whe t, Q $,. (b) (c) Relative miimum:.,.7 or Jue Relative maimum:.,. or Jauar 9 Rt t 7.7 iches Rt t. iches 7 (b) Volume Rt t 7 ( thousa of gallos) Maimum flow: R.7 at t ,.78 is a relative maimum.

30 78 Chapter Itegratio 8. (a) g 9. f (b) g is oegative because the graph of f is positive at the begiig, a geerall has more positive sectios tha egative oes. (c) The poits o g that correspo to the etrema of f are poits of iflectio of g. () No, some zeros of f, like, o ot correspo to a etrema of g. The graph of g cotiues to icrease after because f remais above the -ais. (e) 9. The graph of h is that of g shifte uits owwar. t gt f t f f ht.. False C. True b b si a cos a cos b cos a cosb cos a a b si. False si cos si cos si C si C. Because f is o, f f. The a f f a a a a f a f f. Let u, u i the first itegral. Whe, u. Whe a, u a. a a f f uu f a a a f u u f

31 Sectio. Numerical Itegratio 79 Sectio.. Eact: Trapezoial: Simpso s:. Eact: Trapezoial: Simpso s:. Eact: Trapezoial: Simpso s: 8. Eact: Trapezoial: Simpso s:. Eact: Trapezoial: Simpso s:. Trapezoial: Numerical Itegratio Simpso s:. Graphig utilit:.

32 8 Chapter Itegratio. Trapezoial: si 8 si Simpso s: si 8 si Graphig utilit:.8. Trapezoial: Simpso s: Graphig utilit:. 8. Trapezoial: si si si 7 8 si ta 8 ta ta ta ta ta ta ta ta ta ta ta cos cos 8 cos cos 8.9 Simpso s: cos cos 8 cos cos 8.9 Graphig utilit:.9 si si. Trapezoial: si si Simpso s: Graphig utilit:.8 8 si si si si.8.8. Trapezoial: Liear polomials Simpso s: Quaratic polomials. f f f f f (a) Trapezoial: Error f is maimum i, whe. (b) Simpso s: Error. sice sice f is maimum i, whe.

33 Sectio. Numerical Itegratio 8. f i,. (a) f is maimum whe a f. Trapezoial: Error <., >,.7, > 9.; let. f i, (b) f is maimum whe a f. Simpso s: Error >,., >.7; let. (I Simpso s Rule must be eve.) 8 <., 8. f (a) f i,. 9 (b) f is maimum whe a 8 Trapezoial: Error <., >,8.8, >.7; let. 9 f i, 8 f is maimum whe a f 9. f 8. Simpso s: Error <., >,9.8, >.; let. (I Simpso s Rule must 8 be eve.) 8. f si (a) f si cos i,. is maimum whe a f.8. f Trapezoial: Error.8 <., > 9,.7, > 8.; let 9. (b) f si 8 cos i, f is maimum whe.8 a.8 f 8.8. Simpso s: Error <., >,79., >.; let The program will var epeig upo the computer or programmable calculator that ou use.. f o,. L M R T S

34 8 Chapter Itegratio. f si o,. L M R T S Simpso s Rule: 8 8 si 7.7 si si si 8... si. (a) Trapezoial: f Simpso s: f (b) Usig a graphig utilit, Itegratig,.. Simpso s Rule:.9. Area sq m. The quaratic polomial p passes through the three poits.

CHAPTER 4 Integration

CHAPTER 4 Integration CHAPTER Itegratio Sectio. Atierivatives a Iefiite Itegratio......... 77 Sectio. Area............................. 8 Sectio. Riema Sums a Defiite Itegrals........... 88 Sectio. The Fuametal Theorem of Calculus..........

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