Thomas Calculus 13th [Solutions]

1108 Chapter 15 Multiple Integrals
35. average
2 2
1
a a x 2 2 1
2 a 2 2
2
2 2 2 x y dy dx r dr d a d a
a a x
2
a
a 0 0 3 0 3
36. average
1 2 2 1
2 1
2 2 2
(1 x) y dy dx (1 r cos ) r sin r dr d
0 0
R
2 1 3 2 2
3 2cos 3 2sin
2
1 r 2r cos r dr d 1 d 1
3
0 0 0 4 3 4 3 0 2
37.
2 e 2
2 2 1/2
ln r
e 2ln 2 ln e
2
r dr d r dr d r r r d 1
0 1 r
0 1 0 1 2 e
0 2
1 1 d 2 2 e
38.
39.
2 e 2
ln
2
2ln
2 2
e
r
e
2
dr d r dr d r d d
0 1 r
0 1 r
0 1 0
(ln ) 2
/2 1 cos 2 2
/2 2 3 4
V 2 r
0 1 cos dr d
3 0
3cos 3cos cos d
3
/2
2 15
sin 2 3sin sin
sin 4 4 5
3 8 32 0 3 8
40.
/4 2cos 2 2 4
/4
3/2 3/2
V 4 r
0 0 2 r dr d
3 0
(2 2cos 2 ) 2 d
/4 3 /4
2 2 32 2
2 2 32 cos
6 2 40 2 64
1 cos sin d
cos
3 3 0 3 3 3
0
9
41. (a)
(b)
2 2
/2 2 /2
2
2
x y r
b r
I e dx dy e r dr d lim re dr d
0 0 0 0 0 b 0
/2 2
/2
1 b
lim e 1 d 1 d I
2 0 b
2 0 4 2
t 2 2
x
2e
2 t 2
0 0
lim dt e dt 1, from part (a)
x
2
1
/2
b
r
b
lim
r
lim
1
lim 1
0 0
1 x y
0 0
1 r b 0
1 r
b 1 r 0 b 1 b
42. dx dy dr d dr
2 2
2
2
2 2 2
2 4 2 4
2
4
2 3/2 2
3/2
2 2 3 1 r
1 2
4 1 x y 0 0 1 r
0 2
0
R
2
1 1
2
2 ln d
0 4
(ln 2) d ln 4
0
43. Over the disk x y : dA dr d ln 1 r d
2 2 2
2 2
Over the disk x y 1:
1 dA 2 1 r dr d 2
lim a r dr d
2 2 2 2
1 x y 0 0 1 r 0 a 1 0 1 r
R
2
1 2 1 2
0 1 2 1 2
a
a
2 2
x y
lim ln 1 a d 2 lim ln 1 a 2 , so the integral does not exist over
1
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