08 Chapter 5: Multiple Integrals EXERCISES 5.6 Evaluating Integrals in Clindrical Evaluate the clindrical coordinate integrals in Eercises 6... 3. 4. 5. 6. Changing Order of Integration in Clindrical The integrals we have seen so far suggest that there are preferred orders of integration for clindrical coordinates, but other orders usuall work well and are occasionall easier to evaluate. Evaluate the integrals in Eercises 7 0. 7. 8. 9. 0 0 r 3 8 - r 0 0 r >3 u> 3 + 4r 0 0 0 p u>p 34 - r 0 0 0 0 r > 0 0 -> 0 0 0 +cos u -0 0 0 0 0 3 >3 4 - r - r -4 - r d r dr du > - r sr sin u + d d r dr du r 3 dr d du sr cos u + d r du dr d d r dr du d r dr du d r dr du 3 d r dr du 4r dr du d 0. sr sin u + d r du d dr 0 r - 0. et D be the region bounded below b the plane = 0, above b the sphere + + = 4, and on the sides b the clinder + =. Set up the triple integrals in clindrical coordinates that give the volume of D using the following orders of integration. a. d dr du b. dr d du c. du d dr. et D be the region bounded below b the cone = + and above b the paraboloid = - -. Set up the triple integrals in clindrical coordinates that give the volume of D using the following orders of integration. a. d dr du b. dr d du c. du d dr 3. Give the limits of integration for evaluating the integral as an iterated integral over the region that is bounded below b the plane = 0, on the side b the clinder r = cos u, and on top b the paraboloid = 3r. 4. Convert the integral ƒsr, u, d d r dr du 9 - -0 0 to an equivalent integral in clindrical coordinates and evaluate the result. Finding Iterated Integrals in Clindrical In Eercises 5 0, set up the iterated integral for evaluating 7D ƒsr, u, d d r dr du over the given region D. 5. D is the right circular clinder whose base is the circle r = sin u in the -plane and whose top lies in the plane = 4 -. s + d d d d 4 6. D is the right circular clinder whose base is the circle and whose top lies in the plane = 5 -. r = 3 cos u r sin
5.6 Triple Integrals in Clindrical and Spherical 09 5 0. D is the prism whose base is the triangle in the -plane bounded b the -ais and the lines = and = and whose top lies in the plane = -. r 3 cos 7. D is the solid right clinder whose base is the region in the plane that lies inside the cardioid r = + cos u and outside the circle r = and whose top lies in the plane = 4. 4 r r cos 8. D is the solid right clinder whose base is the region between the circles r = cos u and r = cos u and whose top lies in the plane = 3 -. 3 r cos r cos 9. D is the prism whose base is the triangle in the -plane bounded b the -ais and the lines = and = and whose top lies in the plane = -. Evaluating Integrals in Spherical Evaluate the spherical coordinate integrals in Eercises 6... 3. 4. 5. 6. Changing Order of Integration in Spherical The previous integrals suggest there are preferred orders of integration for spherical coordinates, but other orders are possible and occasionall easier to evaluate. Evaluate the integrals in Eercises 7 30. 7. 8. 9. 0 0 0 p>4 0 0 0 p s - cos fd> 0 0 0 3p> p 0 0 0 p>3 0 0 sec f p>4 sec f 0 0 0 0 -p p>4 p>3 csc f p>6 csc f p p>4 p p 0 0 0 0 p> p/ sin f p> sr cos fd r sin f dr df du 5r 3 sin 3 f dr df du r 3 sin f df du dr 0 r sin 3 f df du dr r sin f dr df du 3r sin f dr df du sr cos fd r sin f dr df du r sin f dr df du r sin f du dr df 30. 5r 4 sin 3 f dr du df p>6 -p/ csc f 3. et D be the region in Eercise. Set up the triple integrals in spherical coordinates that give the volume of D using the following orders of integration. a. dr df du b. df dr du
0 Chapter 5: Multiple Integrals 3. et D be the region bounded below b the cone = + and above b the plane =. Set up the triple integrals in spherical coordinates that give the volume of D using the following orders of integration. a. dr df du b. df dr du 3 Finding Iterated Integrals in Spherical In Eercises 33 38, (a) find the spherical coordinate limits for the integral that calculates the volume of the given solid and (b) then evaluate the integral. 33. The solid between the sphere r = cos f and the hemisphere r =, Ú 0 34. The solid bounded below b the hemisphere r =, Ú 0, and above b the cardioid of revolution r = + cos f cos cos 35. The solid enclosed b the cardioid of revolution r = - cos f 36. The upper portion cut from the solid in Eercise 35 b the plane 37. The solid bounded below b the sphere r = cos f and above b the cone = + Rectangular, Clindrical, and Spherical 39. Set up triple integrals for the volume of the sphere r = in (a) spherical, (b) clindrical, and (c) rectangular coordinates. 40. et D be the region in the first octant that is bounded below b the cone f = p>4 and above b the sphere r = 3. Epress the volume of D as an iterated triple integral in (a) clindrical and (b) spherical coordinates. Then (c) find V. 4. et D be the smaller cap cut from a solid ball of radius units b a plane unit from the center of the sphere. Epress the volume of D as an iterated triple integral in (a) spherical, (b) clindrical, and (c) rectangular coordinates. Then (d) find the volume b evaluating one of the three triple integrals. 4. Epress the moment of inertia I of the solid hemisphere + +, Ú 0, as an iterated integral in (a) clindrical and (b) spherical coordinates. Then (c) find I. Volumes Find the volumes of the solids in Eercises 43 48. 43. 44. 4 4 ( ) ( ) r r 45. 46. cos r 3 cos 38. The solid bounded below b the -plane, on the sides b the sphere r =, and above b the cone f = p>3 r 3 cos
5.6 Triple Integrals in Clindrical and Spherical 47. 48. 3 r sin r cos 49. Sphere and cones Find the volume of the portion of the solid sphere r a that lies between the cones f = p>3 and f = >3. 50. Sphere and half-planes Find the volume of the region cut from the solid sphere r a b the half-planes u = 0 and u = p>6 in the first octant. 5. Sphere and plane Find the volume of the smaller region cut from the solid sphere r b the plane =. 5. Cone and planes Find the volume of the solid enclosed b the cone = + between the planes = and =. 53. Clinder and paraboloid Find the volume of the region bounded below b the plane = 0, laterall b the clinder + =, and above b the paraboloid = +. 54. Clinder and paraboloids Find the volume of the region bounded below b the paraboloid = +, laterall b the clinder + =, and above b the paraboloid = + +. 55. Clinder and cones Find the volume of the solid cut from the thick-walled clinder + b the cones = ; +. 56. Sphere and clinder Find the volume of the region that lies inside the sphere + + = and outside the clinder + =. 57. Clinder and planes Find the volume of the region enclosed b the clinder + = 4 and the planes = 0 and + = 4. 58. Clinder and planes Find the volume of the region enclosed b the clinder + = 4 and the planes = 0 and + + = 4. 59. Region trapped b paraboloids Find the volume of the region bounded above b the paraboloid = 5 - - and below b the paraboloid = 4 + 4. 60. Paraboloid and clinder Find the volume of the region bounded above b the paraboloid = 9 - -, below b the -plane, and ling outside the clinder + =. 6. Clinder and sphere Find the volume of the region cut from the solid clinder + b the sphere + + = 4. 6. Sphere and paraboloid Find the volume of the region bounded above b the sphere + + = and below b the paraboloid = +. Average Values 63. Find the average value of the function ƒsr, u, d = r over the region bounded b the clinder r = between the planes = - and =. 64. Find the average value of the function ƒsr, u, d = r over the solid ball bounded b the sphere r + =. (This is the sphere + + =. ) 65. Find the average value of the function ƒsr, f, ud = r over the solid ball r. 66. Find the average value of the function ƒsr, f, ud = r cos f over the solid upper ball r, 0 f p>. Masses, Moments, and Centroids 67. Center of mass A solid of constant densit is bounded below b the plane = 0, above b the cone = r, r Ú 0, and on the sides b the clinder r =. Find the center of mass. 68. Centroid Find the centroid of the region in the first octant that is bounded above b the cone = +, below b the plane = 0, and on the sides b the clinder + = 4 and the planes = 0 and = 0. 69. Centroid Find the centroid of the solid in Eercise 38. 70. Centroid Find the centroid of the solid bounded above b the sphere r = a and below b the cone f = p>4. 7. Centroid Find the centroid of the region that is bounded above b the surface = r, on the sides b the clinder r = 4, and below b the -plane. 7. Centroid Find the centroid of the region cut from the solid ball r + b the half-planes u = -p>3, r Ú 0, and u = p>3, r Ú 0. 73. Inertia and radius of gration Find the moment of inertia and radius of gration about the -ais of a thick-walled right circular clinder bounded on the inside b the clinder r =, on the outside b the clinder r =, and on the top and bottom b the planes = 4 and = 0. (Take d =. ) 74. Moments of inertia of solid circular clinder Find the moment of inertia of a solid circular clinder of radius and height (a) about the ais of the clinder and (b) about a line through the centroid perpendicular to the ais of the clinder. (Take d =. ) 75. Moment of inertia of solid cone Find the moment of inertia of a right circular cone of base radius and height about an ais through the verte parallel to the base. (Take d =. ) 76. Moment of inertia of solid sphere Find the moment of inertia of a solid sphere of radius a about a diameter. (Take d =. ) 77. Moment of inertia of solid cone Find the moment of inertia of a right circular cone of base radius a and height h about its ais. (Hint: Place the cone with its verte at the origin and its ais along the -ais.) 78. Variable densit A solid is bounded on the top b the paraboloid = r, on the bottom b the plane = 0, and on the sides b
Chapter 5: Multiple Integrals the clinder r =. Find the center of mass and the moment of 84. Mass of planet s atmosphere A spherical planet of radius R inertia and radius of gration about the -ais if the densit is has an atmosphere whose densit is m = m 0 e -ch, where h is the a. dsr, u, d = altitude above the surface of the planet, m 0 is the densit at sea level, and c is a positive constant. Find the mass of the planet s b. dsr, u, d = r. atmosphere. 79. Variable densit A solid is bounded below b the cone = + 85. Densit of center of a planet A planet is in the shape of a and above b the plane =. Find the center of sphere of radius R and total mass M with sphericall smmetric mass and the moment of inertia and radius of gration about the densit distribution that increases linearl as one approaches its -ais if the densit is center. What is the densit at the center of this planet if the densit a. b. dsr, u, d = dsr, u, d =. at its edge (surface) is taken to be ero? 80. Variable densit A solid ball is bounded b the sphere r = a. Theor and Eamples Find the moment of inertia and radius of gration about the -ais 86. Vertical circular clinders in spherical coordinates Find an if the densit is equation of the form r = ƒsfd for the clinder + = a. a. b. dsr, f, ud = r dsr, f, ud = r = r sin f. 87. Vertical planes in clindrical coordinates a. Show that planes perpendicular to the -ais have equations 8. Centroid of solid semiellipsoid Show that the centroid of the of the form r = a sec u in clindrical coordinates. solid semiellipsoid of revolution sr >a d + s >h d, Ú 0, lies on the -ais three-eighths of the wa from the base to the top. b. Show that planes perpendicular to the -ais have equations of the form r = b csc u. The special case h = a gives a solid hemisphere. Thus, the centroid of a solid hemisphere lies on the ais of smmetr three- 88. (Continuation of Eercise 87.) Find an equation of the form r = ƒsud in clindrical coordinates for the plane a + b = c, eighths of the wa from the base to the top. c Z 0. 8. Centroid of solid cone Show that the centroid of a solid right 89. Smmetr What smmetr will ou find in a surface that has circular cone is one-fourth of the wa from the base to the verte. an equation of the form r = ƒsd in clindrical coordinates? Give (In general, the centroid of a solid cone or pramid is one-fourth reasons for our answer. of the wa from the centroid of the base to the verte.) 90. Smmetr What smmetr will ou find in a surface that has 83. Variable densit A solid right circular clinder is bounded b an equation of the form r = ƒsfd in spherical coordinates? Give the clinder r = a and the planes = 0 and = h, h 7 0. Find reasons for our answer. the center of mass and the moment of inertia and radius of gration about the -ais if the densit is dsr, u, d = +.