Thomas Calculus 13th [Solutions]

CHAPTER 15
MULTIPLE INTEGRALS
15.1 DOUBLE AND ITERATED INTEGRALS OVER RECTANGLES
1.
2.
2 4 2 2
4 2
2
2
2xy dy dx xy dx 16x dx 8x
24
1 0 1 0 1
1
2 1 2 2
1 2
2
2
( x y) dy dx xy 1 y dx 2x dx x 4
0 1 0 2 1 0
0
3.
0 1 0 2 1 0 2
0
( x y 1) dx dy x yx x dy (2y 2) dy y 2y
1
1 1 1 2
1 1 1
4.
5.
1 1
1 dx dy x dy dy y
0 0 2 0 6 2 0 6 2 6 6 3
0
0
1 1 2 2 1 3 2 1 2 3
x y x x y
5 y 5 y 2
3 2 2 3 y
3
16 16
3
4 y dy dx 4y dx dx x 16
0 0 0 3 0 3 3
0
0
3 2
6.
2 2 0
3 0 3 3
3 3
2 x y 2 2 2
x y 2xy dy dx xy dx 4x 2x dx 2x
2x
0
0 2 0 2 0 3
2
0
7.
1 1 y
1 1 1
1
dx dy ln|1 xy| dy ln|1 y| dy y ln |1 y| y ln|1 y| 2ln 2 1
0 0 1 xy 0 0 0
0
8.
4 4 4 2
4 4 1/2 3/2
4
x y dx dy 1 x x y dy (4 4 y ) dy 4y 8 y 92
1 0 2 1 4 0 1
3 1 3
9.
10.
11.
ln 2 ln5 2x y ln 2 2x y
ln 5 ln 2 2x 2x 1 2 2 1
ln 2
5 x 1 x
e dy dx e dx 5 e e dx e e 3 (5 e)
0 1 0 1 0
2 2 0 2
1 2 x 1 2
2 1
1
1 x 3 x 3 x 3 x
x y e dy dx x y e dx x e dx x e e 3
0 1 0 2 1 0 2 2 2 0 2
2 /2 2 /2 2
2
2
y sin x dx dy y cos x dy y dy 1 y 3
1 0 1 0 1 2 1 2
12.
2 2 2 2
(sin x cos y) dx dy cos x x cos y dy (2 cos y) dy 2y sin y 2
0 0
13.
4
4 e
2
x e
4
4
ln x (ln x) 1 ln y
dx dy dy dy
1 1 xy 2y 1 1 2y
2 1
1
x
ln 2
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