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.
Area revolved about the
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+
p ! = inner radius
= outer radius
2
2
Volume of the shell
=
Volume of the cylinder - Volume of the hole
=
w%
w%
"
"
!$ p+ ' h(!$ p( ' h
#
&
#
2
2&
=
"
"
w2 %
w2 %
! $ p 2 + pw + ' h ( ! $ p 2 ( pw + ' h
4 &
4 &
#
#
2
2
=
2!phw
To make sense of this formula, you can consider "unrolling" 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$p( x )h( x ) %x
n !"
i =1
n
Volume
=
lim # 2$p( y )h( y ) %y
n !"
i =1
b
= 2! " p(x)h(x)dx
a
b
= 2! " 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! " 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.