Page 236 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 9] MULTIPLE INTEGRALS 227

ðð
2 2 a 2
9:31: If r is the region of Problem 9.30, evaluate e ðx þy Þ dx dy: Ans: ð1 e Þ
r

9.32. By using the transformation x þ y ¼ u; y ¼ uv,show that
ð 1 ð 1 x e 1
e y=ðxþyÞ dy dx ¼
x¼0 y¼0 2

3
3
3
9.33. Find the area of the region bounded by xy ¼ 4; xy ¼ 8; xy ¼ 5; xy ¼ 15. [Hint: Let xy ¼ u; xy ¼ v.]
Ans: 2ln 3
9.34. Show that the volume generated by revolving the region in the first quadrant bounded by the parabolas
2
2
2
2
2
2
y ¼ x; y ¼ 8x; x ¼ y; x ¼ 8y about the x-axis is 279 =2. [Hint: Let y ¼ ux; x ¼ vy.]
3
3
3
3
9.35. Find the area of the region in the first quadrant bounded by y ¼ x ; y ¼ 4x ; x ¼ y ; x ¼ 4y .
Ans: 1
8
ðð
x y sin 1
9.36. Let r be the region bounded by x þ y ¼ 1; x ¼ 0; y ¼ 0. Show that cos dx dy ¼ .[Hint: Let
x y ¼ u; x þ y ¼ v.] x þ y 2
r
TRIPLE INTEGRALS
ð 1 ð 1 ð 2
9.37. (a) Evaluate ffiffiffiffiffiffiffiffiffiffi xyz dz dy dx: ðbÞ Give a physical interpretation to the integral in (a).
p
2
x¼0 y¼0 z¼ x þy 2
Ans: 3
8
ðaÞ
9.38. Find the (a) volume and (b) centroid of the region in the first octant bounded by x=a þ y=b þ z=c ¼ 1,
x
where a; b; c are positive. Ans: ðaÞ abc=6; ðbÞ x ¼ a=4; y ¼ b=4; z ¼ c=4
z
y
9.39. Find the (a) moment of inertia and (b)radius of gyration about the z-axis of the region in Problem 9.38.
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
p
2
2
2
Ans: ðaÞ Mða þ b Þ=10; ðbÞ ða þ b Þ=10
2
2
2
9.40. Find the mass of the region corresponding to x þ y þ z @ 4; x A 0; y A 0; z A 0, if the density is equal
to xyz. Ans: 4=3
2
2
9.41. Find the volume of the region bounded by z ¼ x þ y and z ¼ 2x. Ans: =2
TRANSFORMATION OF TRIPLE INTEGRALS
2
2
9.42. Find the volume of the region bounded by z ¼ 4 x y and the xy plane. Ans: 8
9.43. Find the centroid of the region in Problem 9.42, assuming constant density .
Ans: y z 4
3
x x ¼ y ¼ 0; z ¼
ððð q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
2
9.44. (a) Evaluate x þ y þ z dx dy dz, where r is the region bounded by the plane z ¼ 3 and the cone
r
2
p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2
x þ y .(b)Give a physical interpretation of the integral in (a). [Hint: Perform the integration in
z ¼
ffiffiffi
p
cylindrical coordinates in the order ; z; .] Ans: 27 ð2 2 1Þ=2
p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2
2
2
9.45. Show that the volume of the region bonded by the cone z ¼ x þ y and the paraboloid z ¼ x þ y is =6.
9.46. 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 Ma 2
5