5128_Ch07_pp378-433

Section 7.3 Volumes 407
6. The solid lies between planes perpendicular to the x-axis
at x 1 and x 1. The cross sections perpendicular
to the x-axis between these planes are squares whose diagonals
run from the semicircle y 1 x
2 to the semicircle
y 1 x . 2 8/3
In Exercises 7–10, find the volume of the solid generated by
revolving the shaded region about the given axis.
7. about the x-axis 2p/3 8. about the y-axis 6p
9. about the y-axis 4 p 10. about the x-axis p 2 /16
0
1
1
y
y
x 2y 2
0 2
x tan ⎛– y
⎝ 4
In Exercises 11–20, find the volume of the solid generated by
revolving the region bounded by the lines and curves about the
x-axis.
11. y x 2 , y 0,
32p/5
x 2 12. y x 3 , y 0, x 2 128p/7
13. y 9 x , 2 y 0 36p 14. y x x 2 , y 0 p/30
15. y x, y 1, x 0 2p/3 16. y 2x, y x, x 1 p
20. y x, y 2, x 0 8p
In Exercises 21 and 22, find the volume of the solid generated by
revolving the region about the given line.
21. the region in the first quadrant bounded above by the line
y 2, below by the curve y sec x tan x, and on the left by
the y-axis, about the line y 2 2.301
22. the region in the first quadrant bounded above by the line y 2,
below by the curve y 2 sin x, 0 x p2, and on the left by
the y-axis, about the line y 2 p(3p 8)
In Exercises 23–28, find the volume of the solid generated by
revolving the region about the y-axis.
23. the region enclosed by x 5y 2 , x 0, y 1, y 1 2p
24. the region enclosed by x y 32 , x 0, y 2 4p
⎝
⎛
x
25. the region enclosed by the triangle with vertices 1, 0, 2, 1,
and 1, 1 4p/3
26. the region enclosed by the triangle with vertices 0, 1, 1, 0,
and 1, 1 2p/3
x
2
y
0 3
0
y
x 3y/2
y sin x cos x
17. y x 2 1, y x 3 18. y 4 x 2 , y 2 x 108p/5
117p/5
19. y sec x, y 2, p4 x p4 p 2 2p
x
x
27. the region in the first quadrant bounded above by the parabola
y x 2 , below by the x-axis, and on the right by the line x 2
8p
28. the region bounded above by the curve y x and below by
the line y x 2p/15
Group Activity In Exercises 29–32, find the volume of the solid
described.
29. Find the volume of the solid generated by revolving the region
bounded by y x and the lines y 2 and x 0 about
(a) the x-axis. 8p (b) the y-axis. 32p/5
(c) the line y 2. 8p/3 (d) the line x 4. 224p/15
30. Find the volume of the solid generated by revolving the
triangular region bounded by the lines y 2x, y 0, and
x 1 about
(a) the line x 1. 2p/3 (b) the line x 2. 8p/3
31. Find the volume of the solid generated by revolving the region
bounded by the parabola y x 2 and the line y 1 about
(a) the line y 1. 16p/15 (b) the line y 2. 56p/15
(c) the line y 1. 64p/15
32. By integration, find the volume of the solid generated by
revolving the triangular region with vertices 0, 0, b, 0,
0, h about
(a) the x-axis. (p/3)bh 2 (b) the y-axis. (p/3)b 2 h
In Exercises 33 and 34, use the cylindrical shell method to find the
volume of the solid generated by revolving the shaded region about
the indicated axis.
33. (a) the x-axis 6p/5 (b) the line y 1 4p/5
(c) the line y 85 2p (d) the line y 25 2p
y
1 x 12(y 2 y 3 )
0
34. (a) the x-axis 8p/3 (b) the line y 2 8p/5
(c) the line y 5 8p (d) the line y 58 4p
y
2
0
x y4 — 4
y 2
— 2
y
x 2
— 2
1
1
(2, 2)
2
x
x