A solid of revolution is a solid obtained by rotating a region in the plane about an axis. The sphere and right circular cone are familiar examples of such solids. Each of these is “swept out” as a plane region revolves around an axis.

Volume of Revolution: Disk Method If f (x) is continuous and f (x) ≥ 0 on [a, b], then the solid obtained by rotating the region under the graph about the x-axis has volume [with R = f (x)]

22b b

a a

V R dx f x dx

Calculate the volume V of the solid obtained by rotating the region under y = x2 about the x-axis for 0 ≤ x ≤ 2.

22b b

a a

V R dx f x dx

22 522

0 0

5

5

2 325 5

xx dx

The region between two concentric circles is called an annulus, or more informally, a washer.

2 22 2outer inner

b b

a a

V R R dx f x g x dx

Region Between Two Curves Find the volume V obtained by revolving the region between y = x2 + 4 and y = 2 about the x-axis for 1 ≤ x ≤ 3.

2 2b

a

V f x g x dx

3 3

22 2 4 2

1 1

35 3 5

1

4 2 8 12

8 3 1 812 72 36 125 3 5

2126155 3

V x dx x x dx

x x x

Revolving About a Horizontal Axis Find the volume V of the “wedding band” obtained by rotating the region between the graphs of f (x) = x2 + 2 and g (x) = 4 − x2 about the horizontal line y = − 3.

12 22 2

1

12 22 2

0

4 3 2 3

2 7 5 32

V x x dx

x x dx

2 2b

a

V f x g x dx

Find the volume obtained by rotating the graphs of f (x) = 9 − x2 and y = 12 for 0 ≤ x ≤ 3 about (a) the line y = 12 (b) the line y = 15.

2b

a

V f x dx

3 22

0

12 9 6485

V x dx

3 22 2

0

322

0

11

15 9 3

86 895

V x dx

x dx

Find the volume obtained by rotating the graphs of f (x) = 9 − x2 and y = 12 for 0 ≤ x ≤ 3 about (a) the line y = 12 (b) the line y = 15.

2 2b

a

V f x g x dx

We can use the disk and washer methods for solids of revolution about vertical axes, but it is necessary to describe the graph as a function of y—that is, x = g (y).Revolving About a Vertical Axis Find the volume of the solid obtained by rotating the region under the graph of f (x) = 9 − x2 for 0 ≤ x ≤ 3 about the vertical axis x = −2.

2

outer inner

9 for 0 3 9 (since 0)

9 2, 2

y x x x y x

R y R

2 2d

c

V f y g y dy