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9:31:
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If r is the region of Problem 9.30, evaluate
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By using the transformation x y u; y uv, show that 1 1 x e 1 ey= x y dy dx 2 x 0 y 0 Find the area of the region bounded by xy 4; xy 8; xy3 5; xy3 15. Ans: 2 ln 3 [Hint: Let xy u; xy3 v.]
Show that the volume generated by revolving the region in the rst quadrant bounded by the parabolas y2 x; y2 8x; x2 y; x2 8y about the x-axis is 279=2. [Hint: Let y2 ux; x2 vy.] Find the area of the region in the rst quadrant bounded by y x3 ; y 4x3 ; x y3 ; x 4y3 . Ans: 1 8 Let r be the region bounded by x y 1; x 0; y 0. Show that x y u; x y v.]   x y sin 1 dx dy . [Hint: Let cos x y 2
TRIPLE INTEGRALS 1 1 9.37. (a) Evaluate
x 0 y 0
p xyz dz dy dx: 2 2
x y
b Give a physical interpretation to the integral in (a).
Ans: 9.38. 9.39. 9.40. 9.41.
Find the (a) volume and (b) centroid of the region in the rst octant bounded by x=a y=b z=c 1, " " " where a; b; c are positive. Ans: a abc=6; b x a=4; y b=4; z c=4 Find the (a) moment of inertia p of gyration about the z-axis of the region in Problem 9.38. and (b) radius Ans: a M a2 b2 =10; b a2 b2 =10 Find the mass of the region corresponding to x2 y2 z2 @ 4; x A 0; y A 0; z A 0, if the density is equal to xyz. Ans: 4=3 Find the volume of the region bounded by z x2 y2 and z 2x. Ans: =2
TRANSFORMATION OF TRIPLE INTEGRALS 9.42. 9.43. Find the volume of the region bounded by z 4 x2 y2 and the xy plane. Find the centroid of the region in Problem 9.42, assuming constant density . " " Ans: x y 0; z 4 " 3 (a) Evaluate q x2 y2 z2 dx dy dz, where r is the region bounded by the plane z 3 and the cone Ans: 8
p r z x2 y2 . (b) Give a physical interpretation of the integral in (a). [Hint: Perform the integration in p cylindrical coordinates in the order ; z; .] Ans: 27 2 2 1 =2 9.45. 9.46. Show that the volume of the region bonded by the cone z p x2 y2 and the paraboloid z x2 y2 is =6.
Find the moment of inertia of a right circular cylinder of radius a and height b, about its axis if the density is proportional to the distance from the axis. Ans: 3 Ma2 5
9.47. (a) Evaluate
MULTIPLE INTEGRALS
[CHAP. 9
dx dy dz , where r is the region bounded by the spheres x2 y2 z2 a2 and y2 z2 3=2 (b) Give a physical interpretation of the integral in (a).
x2 y2 z2 b2 where a > b > 0. Ans: a 4 ln a=b 9.48.
(a) Find the volume of the region bounded above by the sphere r 2a cos , and below by the cone  where 0 < < =2. (b) Discuss the case =2. Ans: 4 a3 1 cos4 3 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 b. " " " " " 8 Ans. Taking the z-axis as axis of symmetry: (a) x y 0; z 3 a4 b4 = a3 b3 ; b x y 0, " 8 z 5 a6 b6 = a5 b5
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. Ans: 1 a2 bk 9a2 2b2 , where k constant of proportionality. 6 Find the (a) volume and (b) centroid of the region bounded above by the sphere x2 y2 z2 a2 and below by the plane z b where a > b > 0, assuming constant density. " " " 4 Ans: a 1  2a3 3a2 b b3 ; b x y 0; z 3 a b 2 = 2a b 3 A sphere of radius a has a cylindrical hole of radius b bored from it, the axis of the cylinder coinciding with a diameter of the sphere. Show that the volume of the sphere which remains is 4  a3 a2 b2 3=2 . 3 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). Use Problem 9.53 to nd the volume generated by revolving the circle x2 y b 2 a2 ; b > a > 0 about the x-axis. Ans: 22 a2 b Find the volume of the region bounded by the hyperbolic cylinders xy 1; xy 9; xz 4; xz 36, yz 25, yz 49. [Hint: Let xy u; xz v; yz w:] Ans: 64 Evaluate q 1 x2 =a2 y2 =b2 z2 =c2 dx dy dz, where r is the region interior to the ellipsoid
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