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
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
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
, is rotated about the
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.
the region lies between the lines
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.
To find the points of intersection, we set
and obtain the equation
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
The points of intersection are therefore and
A vertical line segment through this region generates a circular band of
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