Thomas Calculus 13th [Solutions]

Section 6.3 Arc Length 457
9.
dx 1 x dx
2
1 x
2
1 1
2
x
dy x 4 dy x 4 2
x 2 16
L 2 2 2
1 1
2
1 1
2 2
1 x 2 16 1
x 2 16
2
1 x
2 2
dx 1 x dx ln x x
1 x 4 1 x 4 8
1
2
2
y
0.5
2
x
y ln x
8
0 1 2
x
ln 2 4 ln1 1 ln 2 3
8 8 8
0.5
dy
2 2 2
10. 1 dy
x x 1 x 1 1
dx 4x dx 4x 2 2
16x
L 3 1 x 2 1 2
1
1
2
dx
16x
3 2 3 2
x 1 1 1
2 2
dx x dx
1 16x
1 4x
3
x 1 dx x 1 ln x
1 4x
2 4
1
2 3
9 1 ln 3 1 1 ln1 4 1 ln 3
2 4 2 4 4
y
5
2
4
x ln x
y
2 4
3
2
1
0 1 2 3
x
dy
dx
2 1 dy
2
2 1
2
4 1 1
4x dx
4x 2 16x
11. x x x
2 2 4
L 3 1 x 4 1 2
1
1
4
dx
16x
3 2
4 1 1
3 2 1
2 4 2
1 x dx 16x
1
x dx 4x
3 3 3
2 1 1
2
1 x dx x
9 1 1 1
4x
3 4x
53
1
12 3 3 6
y
10
8
3
x 1
y
6
3 4x
4
2
0 1 2 3
x
dy
dx
4 1 dy
2
4 1
2
8 1 1
4x dx
4x 2 16x
12. x x x
4 4 8
L 1 1 x 8 1 2
1
1/2
8
dx
16x
1 2
8 1 1
1 4 1
2 8 4
1/2 x dx 16x
1/2
x dx 4x
1 4
5
x 1 x 1
4
dx
3
1/2 4x
5 12x
1/2
1 1 1 2 373
5 12 160 3 480
1
y
1
5
x
y
5
1
12x
0.5
0 0.5 1 1.5
3
x
13.
dx
dy
2
4 4
sec y 1 dx sec y 1
dy
/4 4 /4 2
L 1 sec y 1 dy sec y dy
/4 /4
/4
tan y 1 ( 1) 2
/4
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