Thomas Calculus 13th [Solutions]

1100 Chapter 15 Multiple Integrals
94. Evaluate the integrals:
2 8
1
8 x
1
3 dx dy
1 y x y 1 1 x y
0.866649
2 2 2 2
3
dy dx
The following graph was generated using
Mathematica.
15.3 AREA BY DOUBLE INTEGRATION
1.
2 2 x 2
dy dx
0 0 0 (2 x ) dx 2 x x
2
2,
0
2 2 y 2
or dx dy
0 0 0 (2 y ) dy 2
2 2
2.
2 4 2 2
2
dy dx (4 2 x) dx 4x x 4,
0 2x 0 0
4 y/2 4 y
or dx dy dy 4
0 0 0 2
3.
2
1 y
1 2
dx dy y y 2 dy
2 y 2 2
3 2 1
y y
3 2
2y
1 1 2 8 2 4 9
3 2 3 2
2
4.
2 3 2
2 y y
2 2 2
dx dy 2y y dy y
0 y
0
4
8 4
3 3
y
3
0
5.
x
ln 2 e ln 2 x x
ln 2
dy dx e dx e
0 0 0 0
2 1 1
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