Math 152 Class Notes
September 15, 2015
7.2 Volume by Cylindrical Shells
Some volume problems are dicult to handle by the method of washers. For instance,
Example 1. Find the volume of the solid obtained by rotating the region bounded by
y = x − x2 and y = 0 about the y -axis.
There is a method, called the method of cylindrical shells, that is easier to use in
the previous example. let S be the solid obtained by rotating the region bounded by
y = f (x), y = 0, x = a and x = b about the y -axis.
Approximate S by cylindrical shells parallel to the y -axis.
The volume of a cylindrical shell can be found by cutting and attening.
The volume of S is
ˆ
b
V =
2πxf (x)dx
a
Similarly, we can rotate a region bounded by x = g(y), x = 0, y = c and y = d about
x-axis. The volume of the resulting solid is
ˆ
V =
d
2πyg(y)dy
c
In general, if a solid of revolution is obtained by rotating a region about a line dierent
from the x-axis or the y -axis, then its volume can be calculated by
ˆ
V =
a
b
2π · [radius] · [height] · [thickness]
Example 1. (Continue) Find the volume of the solid obtained by rotating the region
bounded by y = x − x2 and y = 0 about the y -axis.
Example 2. Find the volume of the solid obtained by rotating the region bonded by
y = x − x2 and y = 0 about x = −2.
Example
√ 3. Find the volume of the solid obtained by rotating the region bounded by
y = 4 − x, x = 0 and y = 0 about the x-axis.
Example
√ 4. Find the volume of the solid obtained by rotating the region bounded by
y = x − 1, y = 0 and x = 5 about y = 3.
Usually, both the methods of washers and shells can be used to nd the volume.
Example 5. Find the volume of the solid obtained by rotating the region bounded by
y = x2 and y = 2x about y = 4.
Example
6. Find the volume of the solid obtained by rotating the region bounded by
√
y = x, y = 0 and x + y = 2 about the x-axis.