Thomas Calculus 13th [Solutions]

1130 Chapter 15 Multiple Integrals
56.
57.
/2 2 2 r
/2 2 2 /2 2
3/2
1
/2
V 8 dz r dr d 8 r 2 r dr d 8 2 r d 8 d 4
0 1 0 0 1 0 3 0 3
1
2 2
2 2 4 r sin 2 2 2 2
V dz r dr d sin
0 0 0 0 0 4 r r sin dr d 8 0
1 d
3
16
58.
2 2 4 r cos r sin 2 2 2 8
2
V dz r dr d
0 0 0 0 0 4 r r (cos sin ) dr d
3 0
(3 cos sin ) d
16
59. The paraboloids intersect when
2 2 2 2 2 2
4x 4y 5 x y x y 1 and z 4
2
/2 1 5 r
/2 1 /2 2 4 1
3 r r
/2
5
2
0 0 4r
0 0 0 2 4
0 0 2
V 4 dz r dr d 4 5r 5r dr d 20 d 5 d
60. The paraboloid intersects the xy -plane when
2 4
3
2 2 2 2
/2 3 9
9 x y 0 x y 9 V 4
dz r dr d
/2 3 3 /2
9 r r
/2
81 17
/2
0 1 0 2 4
1 0 4 4
0
4 9 r r dr d 4 d 4 d 64 d 32
0 1 0
r
2
61.
2 1 4 r
2 1 2
1/2 2 2
3/2
1
8
2 3/2
0 0 0 0 0 0 3 3 0
0
2 1
V 8 dz r dr d 8 r 4 r dr d 8 4 r d 3 8 d
4 8 3 3
3
2 2 2
2 2 2
62. The sphere and paraboloid intersect when x y z 2 and z x y z z
( z 2)( z 1) 0 z 1 or z 2 z 1 since z 0. Thus,
2 2
2
/2 1 2 r
/2 1 2
1/2
3
2
0 0 r
0 0
triple integral V 4 dz r dr d 4 r 2 r r dr d
/2 2
3/2 /2 2 2
8 2 7
1
r
r
d
7
d
0 3 4 0 3 12 6
0
4 1
4 2 4
2 0
x y 1 and the volume is given by the
63. average
1
2 1 1 2 1
2 1 2 1
2
r dz dr d 2r dr d d
2
2 0 0 1 2 0 0 3 0 3
64. average
1
4
3
2
2 1 1 r 2 3
2 1 2 2
r dz dr d r r dr d
2
0 0 1 4
2 1
r
0 0
2
1
3 1 1 2 2
3
2
3
2
sin r
1
r 1 r 1 2r d 0 d d
3
(2 )
3
2 0 8 8 16 0 2 32 0 32 16
0
1
65. average
4
3
2 1 3 3
2
3
2
sin d d d sin d d d
3
0 0 0 16 0 0 8 0 4
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