Thomas Calculus 13th [Solutions]

1096 Chapter 15 Multiple Integrals
73. f ( x, y) dA 1 f 1 , 0 1 f (0, 0) 1 f 1 , 0 1 1 1 0 1 3
4 2 8 8 4 4 2 8 4 32
R
74. f ( x, y) dA 1 f 7 , 11 f 9 , 11 f 7 , 13 f 9 , 13 1 (29 31 33 35) 128 8
4 4 4 4 4 4 4 4 4 16 16
R
75. The ray
Thus,
20 3
9
R
6 meets the circle 2 2
x y 4 at the point ( 3, 1) the ray is represented by the line y
2
3 4 x 2 3 2 2
3 4 x
f ( x, y) dA 4 x dy dx 4 x x 4 x dx 4x
x
0 x/ 3 0 3 3 3 3
2
3/2
0
3
x3 .
76.
1/3 2
2
1 3( y 1)
dy dx dx 3 3 dx 6 dx 6 lim b 1 1 dx
2 0 2 2/3
2
2
2
2 2
x x ( y 1) x x 2 x( x 1) 2 x 1 x
0
x x x x
b
6 lim ln( x 1) ln x b 6 lim ln( b 1) ln b ln 1 ln 2 6 lim ln 1 1 ln 2 6ln 2
b
2
b b
b
77.
3 2 x
V 1 2 x 2 2 1 2 y
0 x
x y dy dx 0
x y 3
dx
x
3 3 3 4
4 1
1 2 7 (2 x) 2 7 (2 x)
2x
x dx x x
0 3 3 3 12 12
0
2 7 1 0 0 16 4
3 12 12 12 3
2 1 1 2 1 2 1 2 2 1
0 x x dx x
y
0 x dy dx 1 y 0 y/ dx dy 1 y 2 y/
dx dy
1 y
78. tan tan
2 2 2
y
1
2 1 y 2 2
2 2
2 1 2
dy dy 1 ln 1 y 2 tan y 1 ln 1 y
0
2
2
2
1 y
1 y 2
0
2
2
1 1 1 2 1
ln 5 2 tan 2 ln 1 4 2 tan 2 1 ln 5
2 2 2
1 1 1
2 ln 5
2 tan 2 2 tan 2 ln 1 4
2 2
79. To maximize the integral, we want the domain to include all points where the integrand is positive and to
exclude all points where the integrand is negative. These criteria are met by the points ( x, y ) such that
2 2
4 x 2y 0 or
2 2
x 2y 4, which is the ellipse
2 2
x 2y 4 together with its interior.
80. To minimize the integral, we want the domain to include all points where the integrand is negative and to
exclude all points where the integrand is positive. These criteria are met by the points ( x, y ) such that
2 2
x y 9 0 or
2 2
x y 9, which is the closed disk of radius 3 centered at the origin.
81. No, it is not possible. By Fubini's theorem, the two orders of integration must give the same result.
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