volution
You have already learned that area is only one of the
many
applications of the definite
integral. Another important application is its use in finding the volume of a three-
dimensional solid. In this section you will study a particular type of three-dimensional
solid--one whose cross sections are similar. Solids of revolution are used commonly
in engineering and manufacturing. Some examples are axles, funnels, pills, bottles,
and pistons, as shown in Figure 7.12.
Solids of revolution
Figure
7.12
If a region in the plane is revolved about a line, the resulting solid is a solid of
revolution, and the line is called the axis of revolution. The simplest such solid is a
right circular cylinder or disk, which is formed by revolving a rectangle about an axis
adjacent to one side of the rectangle, as shown in Figure 7.13. The volume of such a
disk is
Volume of disk -- (area of disk)(width of disk)
= 7rR2w
where R is the radius of the disk and w is the width.
To see how to use the volume of a disk to find the volume of a general solid of
revolution, consider a solid of revolution formed by revolving the plane region in
Figure 7.14 about the indicated axis. To determine the volume of this solid, consider
a representative rectangle in the plane region. When this rectangle is revolved about
the axis of revolution, it generates a representative disk whose volume is
AV = "rrR2Ax.
Approximating the volume of the solid by n such disks of width Ax and radius
R(xi)
produces
Volume of solid
~ 2 7fiR(x;
)]2
Ax
i=1
I1
i=l
~In Figure 7.15, note that you
can determine the variable of integration
by placing a representative rectangle in
the plane region "perpendicular" to the
axis of revolution. If the width of the