Thomas Calculus 13th [Solutions]

Chapter 15 Practice Exercises 1145
b
1
/2 /2
1
/2
dx dy r dr d lim
d lim 1 1 d
1 x y
0 0
1 r
0 b 2 1 r
0 b 2 1 b
R
0
(b)
2 2
2
2
2 2 2 2
1
/2
d
2 0 4
23.
cos( x y z) dx dy dz sin( z y ) sin( z y)
dy dz
0 0 0 0 0
0
cos( z 2 ) cos( z ) cos z cos( z ) dz 0
24.
ln 7 ln 2 ln 5 ( x y z) ln 7 ln 2 ( x y)
ln 7 x
e dz dy dx e dy dx e dx
ln 6 0 ln 4 ln 6 0 ln 6
1
25.
2 2
2 2
4 6
1 x x y 1 x
3 3y
1
(2 x y z)
dz dy dx x dy dx 3x x dx 8
0 0 0 0 0 2 2 0 2 2 35
26. e x z 2 y
dy dz dx e x 1 dz dx e ln x dx x ln x x
e 1
1 1 0
3
z
1 1 z 1 1
27.
V 2 /2 0 2 x /2 0 /2 sin 2
/2
2
0 cos 0 2 0 cos 2 2 0 cos 2 y y
y
dz dx dy y
x dx dy y dy 2 4
0
2
28.
2 2 2 3/2
2 4 x 4 x 2 4 x 2 2 2
V 4 dz dy dx 4 4 x dy dx 4 4 x dx
0 0 0 0 0 0
3/2
2
2 2 1 1
x 4 x 6x 4 x 24 sin x 24sin 1 12
2
0
1 3 1 2 1 3 2 3 1 2
29. average 1 30xz x y dz dy dx 1 15x x y dy dx 1 15x x y dx dy
3 0 0 0 3 0 0 3 0 0
3/2
1
3 2 3 3/2 3/2 5/2 5/2
3
1 1 1 1 5/2 5/2
5 x y dy 5(1 y) 5y dy 2(1 y) 2y
2(4) 2(3) 2
3 0 3 0 3 0 3
0
1 5/2
3 2 31 3
30. average 3
2 a 3
3
2
3
2
sin d d d a sin d d a d 3a
3
4 a 0 0 0 16 0 0 8 0 4
31. (a)
(b)
(c)
2 2 2
2 2 y 4 x y
3 dz dx dy
2 2 2 2
2 y x y
2 /4 2 2
0 0 0 3 sin d d d
2
2
2 2 4 r
2 2 2
1/2
2 2
2
3/2 3
3 dz r dr d 3 r 4 r r dr d 3 1 4 r r d
0 0 r
0 0 0 3 3
0
2 3/2 3/2 3/2
2
2 2 4 d 8 4 2 d 2 8 4 2
0 0
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