Calculus II:
Volumes of Revolution: Shell vs. Disk Method
Revised by Learning Center
Revolution about the y axis
Revolution about the y axis
Shell Method:
 Create differential area parallel to
axis of revolution
 Find the radius of the shell
 Find the height of the shell. It may
be necessary to subtract two
functions
 Find the limits of integration
 Integrate:
Disk Method:
 Create differential area
perpendicular to axis of revolution
 Find the radius of the disk
 If the revolution has a hole,
subtract the volume of the inner
volume by finding the radius of the
inner disk.
 Find the limits of integration
 Integrate:
b
V   2 shell radius shell height dx
b
V    Rdisk  dy
2
a
a
Find the volume of the shape made by the
revolution of y  x and y  x 2 about the y
axis using the shell method.
Find the volume of the shape made by the
revolution of y  x and y  x 2 about the y
axis using the disk method.
b
V   2 Rshell H shell dx
a
1


V   2  x  x  x 2 dx
0
V 

6
b


b


V    Rdisk1 dy    R disk2 dy
2
a
1
V  
0
V 
2
a
 y  dy     y  dy
2
1
2
0

6
Made for the Learning Center: by Lon Farr and Phillip Flanders ©2005
Calculus II:
Volumes of Revolution: Shell vs. Disk Method
Revised by Learning Center
Revolution about the x axis
Revolution about the y axis
Find the volume of the shape made by the
revolution of y  x and y  x 2 about the
x axis using the shell method.
Find the volume of the shape made by the
revolution of y  x and y  x 2 about the
x axis using the disk method.
b
V   2 Rshell H shell dy
a
1
b


b
a


V   2  y  y  y 2 dy
0


V    R disk1 dx    R disk2 dx
2
1
V  
0
2
a
 x  dx    x  dx
2
1
2 2
0
3
V 
10
3
V 
10
Remember
 Make sure your height, radius and
differential are all in terms of the
same variable, but look for the
simplest combination, don’t solve
if you don’t have to.
 You may need to subtract two
functions to find the height.
 Set limits of integration that come
from the area you’re integrating
over, (often, you’ll have to
calculate an intersection.)
Remember
 Make sure to have the radius of the
disk in terms of the same variable
as your differential.
 You may need to subtract two
disks to find a volume with an
internal void
 Set limits of integration that come
from the area you’re integrating
over, (often, you’ll have to
calculate an intersection.)
Made for the Learning Center: by Lon Farr and Phillip Flanders ©2005