# Method of Cylindrical Shells

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

## EXAMPLE 1

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.

# EXAMPLE 2

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

# EXAMPLE 3

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

This document created by Scientific WorkPlace 4.0.