Thomas Calculus 13th [Solutions]

Section 15.2 Double Integrals Over General Regions 1095
65.
V 2 1/ x
2 1/ x 2
1 1
2
1
2
( 1) 1 1 2 1 2 ln
1 1/ x
x dy dx 1 xy y 1/ x dx 1 x x dx 1 x
dx x x 1
2(1 ln 2)
66.
3 sec x
4 /3 sec x 2 /3 /3
3
sec
0 0 1 4 y
V y dy dx y dx
0 3 4 0
sec x x dx
3
0
2 /3
7 ln sec x tan x sec x tan x 2 7 ln 2 3 2 3
3 0 3
67.
68.
1 1
1 ln y
b
dy dx dx x dx lim 1 lim 1 1 1
1 e x y 1 x e 1 x b
x 1 b
b
69. x 3 3 x
3
70.
2
2
1 1/ 1 x
1 1/ 1 x
2 1
1
b
2
1
2
(2y 1) dy dx y y dx dx 4 lim sin x 4 lim [sin b 0]
1 1/ 1 x 1 2 1 2
1/ 1 x 1 x b 1 0 b 1
2
71. 1 1 1
dx dy 2 2 lim tan b tan 0 dy 2 lim b 1 dy
2 2
0
2
0
2
x 1 y 1
y 1 b b y 1
1 1 2
2 lim tan b tan 0 (2 )
b
2
72.
( x 2 y) 2 y x x
b
2 y b b
xe dx dy e lim xe e dy e lim be e
0 0 0 b
0 0 b
1 dy
2 y 1 2b
e dy lim e 1 1
0
2
b
2
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