Thomas Calculus 13th [Solutions]

652 Chapter 8 Techniques of Integration
10.
n 1 n 1 n 1
1 1
1
lim 1 lim n 1 lim 1 1 1 dx sin x
n
n
n 0 0
k 0
n k n
k 0
n k n
k 0
x
2 2 2 2 2 2
1 k
1
1
n
2
11.
dy
dx
dy
2
2
cos 2x 1 1 cos 2x 2cos x;
dx
/4
2 sin t 1
0
/4 2
/4 2
L 1 cos 2t dt 2 cos t dt
0 0
12.
dy
dx
2
2
2
2 1 x 4x
2 4 2 2
2x dy
1 1 2x x 1 x ;
2
2
2
2
2 2
1 x dx
1 x 1 x 1 x
L 1/2 dy
2 1/2 1
2
1
x
0 dx
dx 0 dx 2
1 x
1/2 1/2
2
1/2
1 1 1
1 1
0 1 dx
2
0
1 dx x
1 1 1 ln x
x x 1 x 0 2 ln 3 (0 ln1) ln 3
x
2
13.
b shell shell 1
V 2 radius height dx 2 xy dx
a
0
1
6 x 2
1 x dx;
0
2 2
u 1 x, du dx, x (1 u)
0 2 1 1/2 3/2 5/2
6 (1 u) u du 6 u 2u u du
1 0
3/2 5/2 7/2
1
6 2 u 4 u 2 u
3 5 7 0
6 2 4 2 6 70 84 30 6 16 32
3 5 7 105 105 35
14. V b 2 4 25 dx 4 1 5 1
a y dx 1
2
1
2
x (5 x)
x x 5 x
dx
ln
4
x 5 5 1
5 x x
ln 4 1
4 ln 4
5
15 2 ln 4
4
15.
V
b
2
a
1
shell
radius
shell
height
dx
x x x
1
2 xe dx 2 xe e 2
0 0
16.
ln 2
x
V 2 ln 2 x e 1 dx
0
ln 2 x
x
2 ln 2 e ln 2 xe x dx
0
2 ln 2
x x x
2 ln 2 e ln 2 x xe e x
2
0
2
ln 2
2 2ln 2 ln 2 2ln 2 2 2 ln 2 1
2
2
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