Thomas Calculus 13th [Solutions]

1124 Chapter 15 Multiple Integrals
15.7 TRIPLE INTEGRALS IN CYLINDRICAL AND SPHERICAL COORDINATES
1.
2
2 1 2 r
2 1 2
1/2
2 2
2
3/2
dz r dr d r 2 r r dr d 1 2 r r d
0 0 r
0 0 0 3 3
0
2 3/2 4 2 1
2 2 d
0 3 3 3
3 1
2.
2
3
2 3 18 r
2 3 2
1/2 3 2
2
3/2 4 9 8 2 7
1
2 dz r dr d r 18 r r dr d 18 r r d
0 0 r /3 0 0 3 0 3 12 2
0
3.
3
2 /(2 ) 3 24r 2 /(2 ) 3 2 2 4
/(2 )
dz r dr d 3
0 0 0 0 0 3 r 24 r dr d
0 2 r 6 r d
0
2 4 3 5 2
3
2
4 d 3 17
2 0
2 4 2 2 4
4 16 12 20 0
5
4.
5.
2
/ 3 4 r
/
1 2 2 / 3
z dz r dr d 9 4 r 4 r r dr d 4 4r r dr d
4 r
2
0 0 2
0 0 0 0
4 /
2
2 4
4 2r
r 4 2 d 37
0 0 0 4
4 2 4 15
1/2
2
2 1 2 r
2 1 2
1/2
2 2
2
1/2
3 dz r dr d 3 r 2 r r dr d 3 2 r r d
0 0 r
0 0 0
3
0
3 2
4
0
2 d
3
6 2 8
3 1
6.
2 1 1/2 2 2 2 2 1 3 2 2 2
r sin z dz r dr d r sin r dr d sin 1 d
0 0 1/2 0 0 12 0 4 24 3
7.
2 3 z/3 2 3 4
2
3
r dr dz d z dz d 3 d 3
0 0 0 0 0 324 0 20 10
8.
9.
1 2 1 cos 1 2 2
1
4r dr d dz 2(1 cos ) d dz 6 d 12
1 0 0 1 0 1
2 2
2
1 z 2 2 2 2 1 sin 2 2 1 3 2
cos z r
r z r d dr dz r
z r dr dz z r 2 rz dr dz
0 0 0 0 0 2 4
0 0
0
1 4 z
2 2 1 2
3
3 4
r r z dz z z dz z z
1
0 4
0 0 4 12 4
0
3
10.
2 2
2 4 r 2 2 4 r
2 2
1/2
2
( r sin 1) r d dz dr 2 r dz dr 2 r 4 r r 2r dr
0 r 2 0 0 r 2 0
2
2 2
3/2 3
1 2 8 1 3/2
3 4 r r r
3 2 3 4 3
(4) 8
0
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