Thomas Calculus 13th [Solutions]

412 Chapter 5 Integration
27. f ( y) y, g( y) 2 y, c 1, d 2
d
2
A [ f ( y) g( y)] dy [ y (2 y)]
dy
c
1
2
2 3/2 y
y 2 y dy y 2y
1
3 2
1
4 2 1 4 7 8 2 7
2 4 2 2 2
3 3 2 3 6 6
2 2
28.
2
f ( y) 6 y, g( y) y , c 1, d 2
d
2 2
A [ f ( y) g( y)] dy (6 y y ) dy
c
1
2 3 2
y y
6y
12 2 8 6 1 1
2 3 3 2 3
1
4 7 1 24 14 3 13
3 2 6 6
29.
3 2 2
f ( x) x 3 x x ( x 3)
maximum and f (2) 4 is a minimum.
2
f ( x) 3x 6x 3 x( x 2)
f | | f (0) 0 is a
0 2
A
3 4 3
3 2 3
( x 3 x ) dx x x 81 27 27
0 4
0
4 4
30.
a 1/2 1/2 2 a
1/2 3/2
2 a
A ( a x ) dx ( a 2 ax x)
dx ax 4 ax x
0 0 3 2
0
2 2
2
a 1 4 1 a (6 8 3) a
3 2 6 6
2
a 4 a a a a
3 2
2
31. The area above the x -axis is
5/3 2 1
A 1 2/3
1
0 ( y y)
dy
3y
y 1 ; the area below the x -axis is
5 2 10
0
A 0 2/3
2
1 ( y y)
dy 5/3 2 0
3y
y 11 the
5 2 10
1
total area is A 6
1 A2 5
32.
A
/4 5 /4
(cos x sin x) dx (sin x
0 /4
cos x)
dx
3 /2
(cos
5 /4 sin ) [sin x
/4
cos x] 0 [ cos x
5 /4
sin x]
/4
[sin x
3 /2
cos x] 5 /4
2 2 2 2 2 2
(0 1)
2 2 2 2 2 2
2 2 8 2
( 1 0) 2 4 2 2
2 2 2
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