Thomas Calculus 13th [Solutions]

1116 Chapter 15 Multiple Integrals
33.
2 2 x 4 2 x 2 y 2 2
3 3
dz dy dx
x 3 x y
0 0 (2 x y)/2 0 0 2 2
dy dx
2 2 2
2
3(2 x)
3 2
3 1 x (2 x) (2 x) dx 6 6x 3x
dx
0 2 4 0
2 4
3 3 2
2 (2 x)
3
6x
3 x x
(12 12 4 0) 2 2
2 4 4
0
34.
4 8 8 z
4 8 4 4 2
V dx dy dz (8 2 z) dy dz (8 2 z)(8 z) dz 64 24z 2z dz
0 z z 0 z
0 0
2 3
4
64z 12z 2 z 320
3 0 3
35.
2 2
2 4 x 2 x 2 2 4 x 2 2 2
V 2 dz dy dx 2 ( x 2) dy dx ( x 2) 4 x dx
2 0 0 2 0 2
2 3/2
2
2 2 2 2 2 1 1 2
2 4 x dx x 4 x dx x 4 x 4 sin x 4 x 4 4 4
2 2
2 3 2 2
2 2
36.
2 2 2 2
1 1 y x y 2 1 y
1 3 1 y
2 2 2
V 2 dz dx dy 2 x y dx dy 2 x xy dy
0 0 0 0 0 0 3
0
2 1 2
2
1 2 2 1 2 1 1 2 1 4 2
1 6
0 1 y
3 1 y y dy 2 0 1 y y y dy
3 3 3 3 0
1 y dy
2
7
2 y
1 y
2 6 4
3 7 3 7 7
0
37. average
1
2 2 2 2 2 2 2 2 2
x 9 dz dy dx 1 2x 18 dy dx 1 4x 36 dx 31
8 0 0 0 8 0 0 8 0
3
38. average
1
1 1 2
1
1 1
1
1
( x y z) dz dy dx (2x 2y 2) dy dx (2x 1) dx 0
2 0 0 0 2 0 0 2 0
39. average
1 1 1 2 2 2 1 1 2 2 1
1 2
x y z dz dy dx x y dy dx x 2 dx
0 0 0 0 0 3 0 3
1
40. average
1
2 2 2
1
2 2
1
2
xyz dz dy dx xy dy dx x dx
8 0 0 0 4 0 0 2 0
1
41.
2 2 2
4 1 2 4 cos x 4 2 x/2 4 cos x 4 2 x cos x
4 sin 4 1/2
dx dy dz dy dx dz dx dz z dz
0 0 2 y 2 z 0 0 0 2 z 0 0 z
0 2
1/2
4
(sin 4) z 2 sin 4
0
Copyright
2014 Pearson Education, Inc.