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

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278 D CHAPTER 15 MULTIPLE INTEGRALS 17. In cylindrical coordinates, E is given by {(r, B, z} I 0 ~ B ~ 21r, 0 ~ r ~ 4, - 5 ~ z ~ 4}. So 19. The paraboloid z = 4 - x 2 - y 2 = 4 - r 2 intersects the xy-plane in the circle x 2 + y 2 = 4 or r 2 = 4 =? r = 2, so in cylindrical co~rd inate s, E is given by { (1·, B, z} I 0 ~ (} ~ 1r / 2, 0 ~ r ~ 2, 0 ~ z ~ 4- r 2 }. Thus · JJJE (x + y + z) dV = Io" 12 I; I 0 4 -r 2 (rcos{J + r sin(}+ z) r dzdr d(} = I 0 " 12 I; [r 2 (cos 0 + sin O) z + ~7·z 2 ]~=~-r 2 dr d{J = Io"/ 2 J; [(4r 2 - r 4 )(cos 0 + sin 9) + ~r(4- r· 2?] dr d{J = f '1r / 2 [(1r 3 - l r 5 ) (cos(} +sin 0) - .1.. (4 - r 2 ) 3 ] r = Z d9 .Jo 3 5 12 r=O = I 0 " 12 [~~(cosO + sin 0) + Jf] dO= [ ~(sin 0 - cos B)+ JfB] ; 12 = lfi(1 - 0) + ¥. ~ - ~(0 - 1)- 0 = ~7r + \ 2 5 8 21 . In cylindrical coordinates, E is bounded by the cylinder,. = 1, the plane z = 0, and the cone z = 2r. So E = {(r, 0, z) I 0 ~ 8 ~ 27r, 0 ~ r ~ 1, 0 ~ z ~ 2r} and I I IE x 2 dV = J;'T( f 0 1 J;r r 2 cos 2 9 r dz dr dO = I 0 21f J 0 1 [r 3 cos 2 0 z ] ::~·· dr d9 = I;" Io 1 2r 4 cos 2 8 dr dO = I 02 'T( [%r 5 cos 2 B]~=~ dO= % I 0 2 " cos 2 0 d(} = ~I;"~ (1 +cos 28) dO,;, t [B + t sin 28] ~1f = 2 ; 23. In cylindrical coordinates, E is bounded below by the cone z = r and above by the sphere r 2 + z 2 = 2 or z = v'2 - r 2 . The con~ and the sphere·intersect when 2r 2 = 2 =? r = 1, so E = { (r, 8, z) I 0 ~ 0 ~ 21r, 0 ~ r ~ 1, r ~ z ~ v'2 - r 2 } and the volume is IIIE dV = J;" I0 1 .fr~ rdzdrdO =I;" I; [rzJ~::P drdO = I 0 2 " J 0 1 (r~- r 2 ) drd(} = J 0 21f dO Io 1 (rv'2 - r 2 - r 2 ) dr = 21r [-~(2- r 2 ) 3 1 2 - ~r 3 J: = 27r (- ~) (1 + 1-2 3 1 2 ) = - j7r (2- 2V2) = ~7r (V2 - 1) 25. (a) The paraboloids intersect when x 2 + y 2 = 36 - 3x 2 - 3y 2 =? x 2 + y 2 = 9, so the region of integration is D = {(x,y) I x 2 + y 2 ::; 9}. Then, in cylindrical coordinates, . E = {( r, B, z) I r 2 ~ z ~ 36 - 3r 2 , 0 ~ r ~ 3, 0 ~ (} ~ 27r} and V = g" I; fr 3 26 - 3 r 2 r dz dr d(} = J;" I; (36r - 4r 3 ) dr dB = J;" [18r 2 - r 4 ] ::~dB= J 0 2 " 81 dO= 16271". (b) For constant density K , m = J(V = 1627rJ( from part (a). Since the region is homogeneous and symmetric, Nfyz = Mxz = 0 and © 2012 Ccngoge Learning. All RighiS Resen'Cd. M•l' not be scanned. copkd. or duplieated. or posoc~ oo" publicly aeecssiblc website, in whole or in port.
M xv = I~1f I: I:2 6 - 3 r 2 (zK) r dz dr d() = J( I~ 1f SECTION 15.8 TRIPLE INTEGRALS IN CYLINDRICAL COORDINATES 0 279 I: r [ ~ z 2 ) ; :~~- 3 ,. 2 dr d() =if I 0 2 " I: r((36- 3r 2 ) 2 - r 4 ) dr d() = if I~-:r d8 I 0 3 (8r 5 - 216r 3 + 1296r) dr - K (27r) [.!!.r6 - 216 r4 + 1296 r2] 3 = 11" !((2430) = 243011" J( -2 6 4 ~ 0 . ( - __) ( M 11z Mxz M:xv ) (O O 2430rr Thus x, y, z = --,--,--. = , I
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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 11.9 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 13.2 DERIVATIVES AND INTEGR
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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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184 0 CHAPTER 14 PARTIAL DERIVATIVE
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196 D CHAPTER 14 PARTIAL DERIVATIVE
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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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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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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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