Thomas Calculus 13th [Solutions]

1094 Chapter 15 Multiple Integrals
56.
2/3 2x
1 2 x
xy dA xy dy dx xy dy dx
0 x
2/3 x
R
2/3 2
2x
1 2
2 x
1 xy dx 1 xy dx
0 2 x 2/3 2 x
2/3 3 1 3 1
1 2 1 3
2 x x dx x(2 x)
x dx
0 2 2/3 2 2
2/3
3 3 1 2
x dx
0 2 2/3 2 x x dx
4
2/3
2 3
1
3 x x 2 x 3 16 1 2 4 2 8 6 27 36 16 13
8 0 3 2/3 8 81 3 9 3 27 81 81 81 81 81
57.
3 2 x
3 3 3 4
4 1
V 1 2 x 2 2 1 2 y 1 2 7 (2 ) 2 7 (2 )
2 x x x x x
0 x
x y dy dx 0 x y 3 dx 0
x 3 3 dx 3 12 12
x
0
2 7 1 0 0 16 4
3 12 12 12 3
58.
2
V 2
1 2 x 2 1 2
2 x 1 2 4 3 3 5 4
1
2
2 1 1
2 x
2 x
2 3 5 4
2
2 1 1 16 32 16 40 12 15 320 384 240 189 63
3 5 4 3 5 4 60 60 60 60 60 60 60 20
59.
2 2
V 1 4 x
1 4 1 2 2 2
4 3 ( 4) 4 4 x
x
x dy dx xy y 3x
dx 4
x 4 x 4 4 x 3x 12x
dx
1 3 2 4 3 2
1
x 7x 8x 16 dx 1 x 7 x 4x 16x
1 7 12 64 64 157 1 625
4 4 3 4 4 3 3 3 4 12
60.
61.
2
2 2 4 x
2
V
2 4 x 2 y 2 2
(3 y) dy dx 3y dx 3 4 x 4 x
0 0 0 2 0
2
0
dx
2
2 1
3
3 x 4 x 6 sin x 2x
x 6 4 8 3 16 9 8
2 2 6 2 6 6 3
0
2 3 2 2 2
3 2 2 3
2
V 4 y dx dy 4x y x dy 12 3y dy 12y y 24 8 16
0 0 0 0 0
0
62.
63.
2
2 2 4 x
2
4
2 4 x 2 2 2 y 2
1 2 2 2
V 4 x y dy dx 4 x y dx 4 x dx 8 4x x dx
0 0 0 2 0 2 0
2
0
3 5
2
8x 4 x 1 x 16 32 32 480 320 96 128
3 10 0 3 10 30 15
2 2 x 2 2 3
2 2 3 2 (2 )
12 3 12 x
x
V y dy dx y y dx 24 12 x (2 x) dx 24x 6x
20
0 0 0 0 0
4
0
4 2
64.
V 0 x 1 1 1 x
0 2 1 2
(3 3 ) (3 3 ) 6 1 6 1 4 2 6
1 x 1 x dy dx 0 x 1 x dy dx 1 x dx 0
x dx
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