Page 237 - Schaum's Outline of Theory and Problems of Advanced Calculus

228 MULTIPLE INTEGRALS [CHAP. 9

ððð
dx dy dz
2
2
2
2
9.47. (a) Evaluate , where r is the region bounded by the spheres x þ y þ z ¼ a and
2 2 2 3=2
ðx þ y þ z Þ
r
2
2
2
2
x þ y þ z ¼ b where a > b > 0. (b)Give a physical interpretation of the integral in (a).
Ans: ðaÞ 4 lnða=bÞ
9.48. (a)Find the volume of the region bounded above by the sphere r ¼ 2a cos , and below by the cone ¼
3
4
where 0 < < =2. (b)Discuss the case ¼ =2. Ans: 4 3 a ð1 cos Þ
9.49. Find the centroid of a hemispherical shell having outer radius a and inner radius b if the density (a)is
constant, (b) varies as the square of the distance from the base. Discuss the case a ¼ b.
4
4
3
3
3
x
z
Ans. Taking the z-axis as axis of symmetry: (a) x ¼ y ¼ 0; z ¼ ða b Þ=ða b Þ; ðbÞ x ¼ y ¼ 0,
x
y
y
8
5 6 6 5 5
8
z z ¼ ða b Þ=ða b Þ
MISCELLANEOUS PROBLEMS
9.50. Find the mass of a right circular cylinder of radius a and height b if the density varies as the square of the
distance from a point on the circumference of the base.
2
2
2
Ans: 1 a bkð9a þ 2b Þ, where k ¼ constant of proportionality.
6
2
2
2
2
9.51. Find the (a) volume and (b) centroid of the region bounded above by the sphere x þ y þ z ¼ a and
below by the plane z ¼ b where a > b > 0, assuming constant density.
3
2
3
Ans: 1 ð2a 3a b þ b Þ; x y z 3 2
3 ðbÞ x ¼ y ¼ 0; z ¼ ða þ bÞ =ð2a þ bÞ
4
ðaÞ
9.52. Asphere of radius a has a cylindrical hole of radius b bored from it, the axis of the cylinder coinciding with a
4 3 2 2 3=2 Š.
3
diameter of the sphere. Show that the volume of the sphere which remains is ½a ða b Þ
9.53. A simple closed curve in a plane is revolved about an axis in the plane which does not intersect the curve.
Prove that the volume generated is equal to the area bounded by the curve multiplied by the distance
traveled by the centroid of the area (Pappus’ theorem).
2
2
2
9.54. Use Problem 9.53 to find the volume generated by revolving the circle x þðy bÞ ¼ a ; b > a > 0 about
2 2
the x-axis. Ans: 2 a b
9.55. Find the volume of the region bounded by the hyperbolic cylinders xy ¼ 1; xy ¼ 9; xz ¼ 4; xz ¼ 36, yz ¼ 25,
yz ¼ 49. [Hint: Let xy ¼ u; xz ¼ v; yz ¼ w:] Ans: 64
ðð ð q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
2
2
2
2
9.56. Evaluate 1 ðx =a þ y =b þ z =c Þ dx dy dz, where r is the region interior to the ellipsoid
r
2
2
2
2
2
2
x =a þ y =b þ z =c ¼ 1. [Hint: Let x ¼ au; y ¼ bv; z ¼ cw. Then use spherical coordinates.]
2
Ans: 1 abc
4
ðð
2 2 2
2
2
9.57. If r is the region x þ xy þ y @ 1, prove that e ðx þxyþy Þ dx dy ¼ p ðe 1Þ. [Hint: Let
ffiffiffi
e 3
r
x ¼ u cos v sin , y ¼ u sin þ v cos and choose so as to eliminate the xy term in the integrand.
Then let u ¼ a cos , v ¼ b sin where a and b are appropriately chosen.]
x
ð ð x ð x 1 ð x
n
9.58. Prove that FðxÞ dx ¼ ðx uÞ n 1 FðuÞ du for n ¼ 1; 2; 3; ... (see Problem 9.22).
0 0 0 ðn 1Þ! 0