# 7.3 Shell Method

```7.3 Shell Method
The Shell Method provides another method for determining the volume of a solid of revolution. We determine
the volume by dividing the solid into a series of concentric shells centered about the axis of revolution. By
letting the number of shells go to infinity, we determine the volume of the solid with an integral.
y-axis
Let
Series of concentric
shells
Representative shell
w = width of the representative shell
p = radius to the center of the cylindrical section
w
w
p+
2
2
Volume of the shell
=
Volume of the cylinder - Volume of the hole
=
w%
w%
&quot;
&quot;
!\$ p+ ' h(!\$ p( ' h
#
&amp;
#
2
2&amp;
=
&quot;
&quot;
w2 %
w2 %
! \$ p 2 + pw + ' h ( ! \$ p 2 ( pw + ' h
4 &amp;
4 &amp;
#
#
2
2
=
2!phw
To make sense of this formula, you can consider &quot;unrolling&quot; the representative shell into the following shape.
w
h
2!p
The volume of the solid is determined by taking the limit of the sum of the volumes of n shells as n goes to
infinity as follows:
n
Volume
=
lim # 2\$p( x )h( x ) %x
n !&quot;
i =1
n
Volume
=
lim # 2\$p( y )h( y ) %y
n !&quot;
i =1
b
= 2! &quot; p(x)h(x)dx
a
b
= 2! &quot; p(y)h(y)dy
a
Example 1
Determine the volume of revolution formed by revolving the area bounded by
y = x 2 + 1, y = 0, x = 0, and x = 1 about the y-axis. Use the Shell Method to determine the volume.
b
V = 2! &quot; p(x)h(x)dx
a
Example 2
Determine the volume of the solid of revolution generated by revolving the area between the functions
y = 4 + 2 x , x = 0 and y = x 3 about the y-axis. Use the Shell Method.
```