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SURFACE INTEGRALS 10.52. (a) Evaluate x2 y2 dS, where S is the surface of the cone z2 3 x2 y2 bounded by z 0 and z 3.

(b) Interpret physically the result in (a).

Ans:

10.53. Determine the surface area of the plane 2x y 2z 16 cut o by (b) x 0; y 0, and x2 y2 64. Ans: a 9; b 24

10.54. Find the surface area of the paraboloid 2z x2 y2 which is outside the cone z p Ans: 2  5 5 1 3

10.55. Find the area of the surface of the cone z2 3 x2 y2 cut out by the paraboloid z x2 y2 . Ans: 6 10.56. Find the surface area of the region common to the intersecting cylinders x2 y2 a2 and x2 z2 a2 . Ans: 16a2 p 10.57. (a) Obtain the surface area of the sphere x2 y2 z2 a2 contained within the cone z tan  x2 y2 , 0 <  < =2. (b) Use the result in (a) to nd the surface area of a hemisphere. (c) Explain why formally placing   in the result of (a) yields the total surface area of a sphere. Ans: a 2a2 1 cos  ; b 2a2 (consider the limit as  ! =2 10.58. Determine the moment of inertia of the surface of a sphere of radius a about a point on the surface. Assume a constant density . Ans: 2Ma2 , where mass M 4a2  10.59. (a) Find p the centroid of the surface of the sphere x2 y2 z2 a2 contained within the cone z tan  x2 y2 , 0 <  < =2. (b) From the result in (a) obtain the centroid of the surface of a hemisphere. Ans: a 1 a 1 cos  ; b a=2 2 THE DIVERGENCE THEOREM 10.60. Verify the divergence theorem for A 2xy z i y2 j x 3y k taken over the region bounded by 2x 2y z 6; x 0; y 0; z 0. Ans: common value 27 10.61. Evaluate

A n dS, where A 2x 3z i xz y j y2 2z k and S is the surface of the sphere having Ans: 108

x dy dz y dz dx z dx dy, where S is the surface of the region bounded by the (a) by using the divergence theorem, (b) directly.

4xz dy dz y2 dz dx yz dx dy, where S is the surface of the cube bounded by x 0, y 0, (a) directly, (b) By Green s theorem in space (divergence theorem).

n dS 0, where n is the outward drawn normal to any closed surface S. [Hint: Let A c, Express the divergence theorem in this special case. Use the

10.67. If n is the unit outward drawn normal to any closed surface S bounding the region V, prove that div n dV S

STOKES THEOREM 10.68. Verify Stokes theorem for A 2yi 3xj z2 k, where S is the upper half surface of the sphere x2 y2 z2 9 and C is its boundary. Ans. common value 9 10.69. Verify Stokes theorem for A y z i xzj y2 k, where S is the surface of the region in the rst octant bounded by 2x z 6 and y 2 which is not included in the (a) xy plane, (b) plane y 2, (c) plane 2x z 6 and C is the corresponding boundary. Ans. The common value is (a) 6; b 9; c 18 10.70. Evaluate z 2 r A n dS, where A x z i x3 yz j 3xy2 k and S is the surface of the cone Ans: 12 10.71. If V is a region bounded by a closed surface S and B r A, prove that