Exercicios resolvidos James Stewart vol. 2 7ª ed - ingles

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260 D CHAPTER 15 MULTIPLE INTEGRALS 29. / 3 {~ sin(x 2 + y 2 )dy dx = {" r sin e) r dr dB -3lo · Jo Jo = fo" dB J; r sin (r 2 ) dr = [B) ~ [-~cos (r 2 )J~ -3 0 3 X = 1r (-~) (cos9 -1) = ~ (1 - cos9) 31. y f 0 " 14 fov'2 (r cos B + r sin fJ) r dr dB == f 0 " 14 (cos B + sin B) dB f 0 v'2 1·2 dr == [sin(}- cosB ) ~/ 4 [ i r 3 ]~ = ( ¥ - ¥-O+ 1] . i(2v'2- o) = ¥ 33. D = {( r-, B) [ 0::; r ::; 1, 0 ::; 8::; 271' }, so ff D e
SECTION 15.5 APPLICATIONS OF DOUBLE INTEGRALS 0 261 ·Y ?r/4 2 = r r 1rr/4 [ r 3 4 ] r=2 cos 8 sinfJdrd8= .!:.... cos8sin8 d() Jo J 1 o 4 r = 1 151rr/ 4 . 15 [sin 2 ()] .,.; 4 15 = - sm8cos9d(J = - -- = - 4 0 4 2 0 16 • . 2 2 41. (a) We integrate by parts with u = x and dv =·xe-x dx. Then du = dx and v = - ~e-x , so f "" x 2 e- "' 2 dx = lim rt x 2 e-"' 2 dx = lim ( - l xe- "' )t 2 + rt l e- "' 2 dx) Jo t-+oo Jo t-+oo 2 0 Jo 2 = lim (_.!.te-t 2) + .!. roo e-x 2 dx = 0 + .!. roo e- "' 2 dx t-+oo 2 2 Jo 2 Jo [by !'Hospital's Rule] 1 f "" _.,2 d = 4 -oo e X [by Exercise 40(c)] 2 [since e-"' is an even function] (b) Let u = ..JX. Then u 2 = x .::? dx = 2udu .::? 00 f. 0 .../Xe- "'dx = lim I; .../Xe-"'dx = lim I 0 ,/f.ue-" 2 00 2u du = 2I 0 u 2 e- u 2 du = 2 U.J7T) · t-oo # t-oo [by part(a)] = ~ft. 15.5 Applications of Double Integrals 1. Q = IIv O"(x, y) dA = I~ I: (2x + 4y) dydx =I~ [2xy + 2y 2 ] ~=~ dx = I~ (lOx + 50- 4x- 8) dx = J~ (6x + 42) dx = [3x 2 + 42x]~ = 75 + 210 = 285 C 3. m = IIv p(x, y) dA = I 1 3 I 1 4 ky 2 dydx = k I 1 3 dx .{ 1 4 y 2 dy = . k [xJ ~ [h 3 ]~ = k(2)(21) = 42k, x = ~ IIv xp(x,y) dA = 4 ~k I : I 1 4 kxy 2 dydx = 4 1 2 I 1 3 xdx I 1 4 y 2 dy = f2 [ tx 2 ]~ [k y 3 ]~ = 4\(4)(21) = 2, - 1 ff ( )dA - 1 f a f 4k a d dx- 1 f sdx r4 ad _ 1 [ ]3 [1 4] 4 - 1 ( 2 )( 2 11:;) 85 Y = m D yp x , Y - 42k 1 . 1 Y y - 42 1 Jt y y - 42 X 1 4Y 1 - 42 -4- = 28 Hence m = 42k, (x, Y) = (2, ~) . r2f3-x( ) r2 [ 1 2]v=3-:z: dx r2 [ ( a ) 1 ( 3 )2 1 2] 5. m= Jo .,12 x+y dydx = Jo xy+2Y v=x/ 2 =J~ x 3- 2x + 2 - x - 8x dx = J 0 2 (-~x 2 + ~) dx = [- ~(tx 3 ) + ~x ] ~ = 5, . r2J·3-x( 2 )d dx r2 ( 2 1 2]11=8- :z: dx r2 (9 9 3) d 9 lvfv = Jo :r:/ 2 x + xy y = Jo x y + 2xy 1l=x/2 = Jo 2x - gX x = 2• lvfx = Jo f2J3-v( ., 2)d d r2.[1 2 1 3)11=3-x d r 2 (g 9 )dx 12 xy + y y x = Jo 2xy + 3Y 9 11 =, 12 x = Jo - 2x = . --) (lvfy Mx) (3 3) Hencem = 6, ( x_, y = m '--;:; = 4' 2 . 7. m = J~ 1 J~ -"' 2 kydydx = k f~ 1 [ ~y 2 ]~=~-"' 2 dx = ~k f~ 1 (1 - x 2 ? dx = ~ k f~ 1 (1 - 2x2 + x 4 ) dx = ~k [x- ix 3 + %x 5 ] ~ 1 = ~k (1 - i + -k + 1- % + %) = 1 85 k, ® 2012 Cengnge Learning. All Rights Reserved. Mny not tJe scanned, copied, or duplicated, or posted to a publicly acc•:ssiblc website, in whole or in pan.
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- STUDENT SOLUTIONS MANUAL for STEW
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.. BROOKS/COLE ~ I ~~r CENGAGE Lear
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D ABBREVIATIONS AND SYMBOLS CD cu D
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viii o CONTENTS 12.4 The Cross Prod
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10 D PARAMETRIC EQUATIONS AND POLAR
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SECTION 10.1 CURVES DEFINED BY PARA
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SECTION 10.1 CURVES DEFINED BY PARA
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SECTION 10.2 CALCULUS WITH PARAMETR
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SECTION 10.3 POLAR COORDINATES 0 13
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SECTION 10 .~ POLAR COORDINATES 0 1
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SECTION 10.4 A~~S AND LENGTHS IN PO
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SECTION 10.4 AREAS AND LENGTHS IN P
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SECTION 10.4 AREAS AND LENGTHS IN P
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SECTION 10.5 CONIC SECTIONS 0 27 5.
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SECTION 10.5 CONIC SECTIONS 0 29 35
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x2 y2 y2 a:2 _ a2 b 61. ;_2 - - = 1
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SECTION 10.6 CONIC SECTIONS IN POLA
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CHAPTER 10 REVIEW 0 35 the length o
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CHAPTER 10 REVIEW 0 37 EXERCISES 1.
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CHAPTER 10 REVIEW 0 39 25. x = t +
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CHAPTER 10 REVIEW 0 41 2 2 . 45. ~
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0 PROBLEMS PLUS l lt sin u dx cost
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11 . D INFINITE SEQUENCES AND SERIE
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SECTION 11.1 SEQUENCES 0 47 35. a,.
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SECTION 11.1 SEQUENCES D 49 71. Sin
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SECTION 11.2 SERIES 0 51 ak+l- a1.:
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SECTION 11.2 SERIES 0 53 13. n s.,
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SECTION 11.2 SERIES 0 55 47. F~r th
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. . (1+ c)- 2 scn es and set 1t equ
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SECTION 11.3 THE INTEGRAL TEST AND
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, SECTION 11.3 THE INTEGRAL TEST AN
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SECTION 11.4 THE COMPARISON TESTS D
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SECTION 11.5 ALTERNATING SERIES 0 6
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SECTION 11.5 ALTERNATING SERIES D 6
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SECTION 11.6 ABSOLUTE CONVERGENCE A
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SECTION 11.6 ABSOLUTE CONVERGENCE A
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17 lim I an+l I= SECTION 11.7 STRAT
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SECTION 11.8 POWER SERIES 0 75 l 00
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SECTION 11.8 POWER SERIES 0 n (b) I
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SECTION 11.9 REPRESENTATIONS OF FUN
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SECTION 11.9 REPRESENTATIONS OF FUN
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SECTION 11.10 TAYLOR AND MACLAURIN
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61 _ x_ x · sin x - x- tx 3 + 1~
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SECTION 11.11 APPLICATIONS OF TAYLO
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CHAPTER 11 REVIEW 0 97 J'(xn)(xn -
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CHAPTER 11 REVIEW 0 99 oo f (n) (O)
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CHAPTER 11 REVIEW 0 101 l 23. Consi
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49./- 1 - dx = -ln{4- x) + C and 4-
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D PROBLEMS PLUS 1. It would be far
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CHAPTER 11 PROBLEMS PLUS D 107 l 9.
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CHAPTER 11 PROBLEMS PLUS 0 109 x x
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112 0 CHAPTER 12 VECTORS AND THE GE
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120 D CHAPTER 12 VECTORS AND THE GE
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136 D CHA~TER 12 VECTORS AND THE GE
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D PROBLEMS PLUS 1. Since three-dime
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l CHAPTER 12 PROBLEMS PLUS 0 149 Eq
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13 D VECTOR FUNCTIONS 13.1 Vector F
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SECTION 13.1 VECTOR FUNCTIONS AND S
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SECTION 13.1 VECTOR FUNCTIONS AND S
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SECTION 13.2 DERIVATIVES AND INTEGR
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SECTION 13.2 DERIVATIVES AND INTEGR
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SECTION 13.3 ARC LENGTH AND CURVATU
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SECTION 13.4 MOTION IN SPACE: VELOC
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2 SECTION 13.4 MOTION IN SPACE: VEL
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CHAPTER 13 REVIEW D 173 43. The tan
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5. f~(t 2 i +tcos 1rtj +sin 1rt k )
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CHAPTER 13 REVIEW 0 177 23. (a) r (
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180 0 CHAPTER 13 PROBLEMS PLUS (-2a
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182 0 CHAPTER 13 PROBLEMS PLUS vect
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184 0 CHAPTER 14 PARTIAL DERIVATIVE
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196 D CHAPTER 14 PARTIAL DERIVATIVE
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SECTION 16.4 GREEN'S THEOREM D 313
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SECTION 16.4 GREEN'S THEOREM 0 315
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8 8 8 (b)clivF = 'V ·iF=- (x+yz) +
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SECTION 16.5 CURL AND DIVERGENCE 0
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SECTION 16.6 PARAMETRIC SURFACES AN
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that is, D = {( x, y) I x 2 + y 2 :
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SECTION 16.7 SURFACE INTEGRALS 0 32
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SECTION 16.8 STOKES' THEOREM 0 333
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dS SECTION 16.9 THE DIVERGENCE THEO
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CHAPTER 16 REVIEW 0 337 27. JI 5 cu
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CHAPTER 16 REVIEW 0 339 TRUE-FALSE
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CHAPTER 16 REVIEW D 341 Alternate s
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344 0 CHAPTER 16 PROBLEMS PLUS Simi
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3.c6 0 CHAPTER 17 SECOND-ORDER DIFF
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348 0 CHAPTER 17 SECOND-ORDER DIFFE
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352 D CHAPTER 17 SECOND-ORDER DIFFE
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356 D CHAPTER 17 SECOND-ORDER DIFFE
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0 APPENDIX Appendix H Complex Numbe
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APPENDIX H COMPLEX NUMBERS 0 361 43
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Exercicios resolvidos James Stewart vol. 2 7ª ed - ingles

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