Page 270 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 10] LINE INTEGRALS, SURFACE INTEGRALS, AND INTEGRAL THEOREMS 261

SURFACE INTEGRALS
ðð
2
2
2
2
2
10.52. (a) Evaluate ðx þ y Þ dS, where S is the surface of the cone z ¼ 3ðx þ y Þ bounded by z ¼ 0 and z ¼ 3.
S
(b)Interpret physically the result in (a). Ans: ðaÞ 9
10.53. Determine the surface area of the plane 2x þ y þ 2z ¼ 16 cut off by (a) x ¼ 0; y ¼ 0; x ¼ 2; y ¼ 3,
2
2
(b) x ¼ 0; y ¼ 0, and x þ y ¼ 64. Ans: ðaÞ 9; ðbÞ 24
2 2 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2
2
10.54. Find the surface area of the paraboloid 2z ¼ x þ y which is outside the cone z ¼ x þ y .
ffiffiffi
Ans: 2
p
3 ð5 5 1Þ
2
2
2
2
2
10.55. Find the area of the surface of the cone z ¼ 3ðx þ y Þ cut out by the paraboloid z ¼ x þ y .
Ans: 6
2
2
2
2
2
2
10.56. Find the surface area of the region common to the intersecting cylinders x þ y ¼ a and x þ z ¼ a .
Ans: 16a 2
2 2 2 2 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2
2
10.57. (a) Obtain the surface area of the sphere x þ y þ z ¼ a contained within the cone z tan ¼ x þ y ,
0 < < =2. (b)Use the result in (a)to find the surface area of a hemisphere. (c) Explain why formally
placing ¼ in the result of (a)yields the total surface area of a sphere.
2
2
Ans: ðaÞ 2 a ð1 cos Þ; ðbÞ 2 a (consider the limit as ! =2Þ
10.58. Determine the moment of inertia of the surface of a sphere of radius a about a point on the surface. Assume
2
2
aconstant density . Ans: 2Ma , where mass M ¼ 4 a
2
2
2
10.59. (a)Find the centroid of the surface of the sphere x þ y þ z ¼ a 2 contained within the cone
p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
x þ y ,0 < < =2. (b)From the result in (a) obtain the centroid of the surface of a hemi-
z tan ¼
sphere. Ans: 1 að1 þ cos Þ; ðbÞ a=2
2
ðaÞ
THE DIVERGENCE THEOREM
2
10.60. Verify the divergence theorem for A ¼ð2xy þ zÞi þ y j ðx þ 3yÞk taken over the region bounded by
2x þ 2y þ z ¼ 6; x ¼ 0; y ¼ 0; z ¼ 0. Ans: common value ¼ 27
ðð
2
10.61. Evaluate F n dS, where F ¼ðz xÞi xyj þ 3zk and S is the surface of the region bounded by
S
2
z ¼ 4 y ; x ¼ 0; x ¼ 3 and the xy plane. Ans. 16
ðð
2
10.62. Evaluate A n dS, where A ¼ð2x þ 3zÞi ðxz þ yÞj þð y þ 2zÞk and S is the surface of the sphere having
S
center at ð3; 1; 2Þ and radius 3. Ans: 108
ðð
10.63. Determine the value of xdy dz þ ydz dx þ zdxdy, where S is the surface of the region bounded by the
S
2
2
cylinder x þ y ¼ 9 and the planes z ¼ 0 and z ¼ 3, (a)by using the divergence theorem, (b)directly.
Ans: 81
ðð
2
10.64. Evaluate 4xz dy dz y dz dx þ yz dx dy, where S is the surface of the cube bounded by x ¼ 0, y ¼ 0,
S
z ¼ 0, x ¼ 1; y ¼ 1; z ¼ 1, (a)directly, (b)ByGreen’s theorem in space (divergence theorem).
Ans. 3/2
ðð
10.65. Prove that ðr AÞ n dS ¼ 0for any closed surface S.
S