In addition to the slicing methods, there is another way to calculate the volume of a solid of revolution. While the slicing method is based on the idea of approximating cross sections taken perpendicular to the axis of rotation, the method of cylindrical shells uses approximating hollow cylinders centered about the axis of rotation.

To describe this method, we let be the region bounded by the graph of a continuous nonnegative function and the -axis for . If the region is rotated about the -axis, a solid is generated.

To calculate the volume
of
,
we partition the interval
using the equal-length partition

where
Then we choose the numbers
to be the midpoints of the subintervals. That is,
Using these choices, we approximate the region
by rectangles with base
and height
. If we rotate one of these rectangles about the
-axis,
we obtain a cylindrical shell. Combining all such shells, we obtain a solid
whose volume approximates the volume of

Consider the jth shell. Its height is
and the area of its base is
Consequently, its volume
is

Thus, by summing the volumes of all of the shells, we approximate the volume
by

To see that the right-hand side of this approximation is a Riemann sum, we
factor

and rewrite the approximation as

Since
we have

and taking the limit as
these Riemann sums approach

Assuming that these sums also limit to
we obtain the following volume formula for

If the region
bounded by the graph of the continuous nonnegative function
and the
-axis,
for
, is rotated about the
-axis,
the volume
of the resulting solid is

The region bounded by the graph of and the -axis is rotated about the -axis. Find the volume of the resulting solid.

**Solution**

Since
the region lies between the lines
and

Equation

can be generalized to regions bounded below by curves other than the
-axis,
as the following example shows.

The region is bounded by the graphs of and . Find the volume of the solid generated by revolving about the -axis.

**Solution**

To find the points of intersection, we set
and obtain the equation

or

so
or
When the interval
is partitioned, the approximating rectangles are bounded above by
and below by
Since the factor
represents the height of the approximating rectangles, we modify equation to
the following:

**REMARK ** We may state the generalization of equation

observed in Example 2 as follows: If the region
bounded above by the graph of
and below by the graph of
for
is rotated about the
-axis,
the volume of the resulting solid is

Find the volume of the solid obtained by revolving about the line the region bounded by the graphs of and

**Solution** The region
that is bounded above by the graph of
and below by the graph of
To find the points of intersection of these two graphs we set
which gives the equation

or

The points of intersection are therefore
and

A vertical line segment through this region generates a circular band of
height

Since
the axis of rotation is the vertical line
the radius of this circular band is
We must therefore replace the factor
in the integrand in formula

by
the factor
The volume is therefore