VOLUMES OF SOLIDS OF
REVOLUTION
• Questions involving the area of a region
between curves, and the volume of a solid
formed when this region is rotated about a
horizontal or vertical line, appear regularly on
both the AP Calculus AB and BC exams.
• Students have difficulty when the solid is
formed by use a line of rotation other than the
x- or y-axis.
• These types of volume are part of the type of
volume problems students must solve on the
AP test.
• Students should find the volume of a solid
with a known cross section.
• The Shell method is not part of the AB or the
BC course of study anymore.
• The four examples in the Curriculum Module
use the disk method or the washer method.
Example 1 Line of Rotation Below the
Region to be Rotated
• Picture the solid (with a hole)
generated when the region
bounded by y  x  2
and y  e x are revolved
about the line y = -2.
• First find the described region
• Then create the reflection
over the line y=-2
1
-1
1
-1
-2
-3
-4
-5
-6
Example 1
• Think about each of
the lines spinning and
creating the solid.
• Draw one
representative disk.
• Draw in the radius.
Example 1
• Find the radius of the
larger circle, its area
and the volume of the
disk.
r  2  y o u tsid e
A re a    2  y o u tsid e 
2
2
V o lu m e    2  y o u tsid e   x

  2 
x 2

2
x
• Sum up these cylinders
to find the total
volume
n
V o lu m e 

k 1
2 
x 2

2
x
The larger the number of disks and the thinner each disk, the
smoother the stack of disks will be. To obtain a perfectly
smooth solid, we let n approach infinity and Δx approach 0.
• The points of intersection can
be found using the calculator.
• Store these in the graphing
calculator
(A=-1.980974,B=0.13793483)
(C=0.44754216,D=1.5644623)
• Write an integral to find the
volume of the solid.
C
V o lu m e 

A
2 
x 2
 dx
2
 7 1 .9 8 3 3 3 6 3
Example 1
• Find the radius of the
smaller circle, its area
and the volume of the
disk.
r  2  y in sid e
A re a    2  y in sid e 
2
2
V o lu m e    2  y in sid e   x

  2  e
x

2
x
• Sum up these cylinders
to find the total
volume
n
V o lu m e 
  2  e 
x
2
x
k 1
The larger the number of disks and the thinner each disk, the
smoother the stack of disks will be. To obtain a perfectly
smooth solid, we let n approach infinity and Δx approach 0.
• Using the points of
intersection write a second
integral for the inside volume.
(A=-1.980974,B=0.13793483)
(C=0.44754216,D=1.5644623)
C
V o lu m e 
  2  e 
x
A
2
d x  5 2 .2 5 8 6 1 0
Example 1
• The final volume will
be the difference
between the two
volumes.
C

A
2 
x 2
C
 d x    2  e  d x
2
x
A
2
 1 9 .7 2 4 o r 1 9 .7 2 5
Example 2 Line of Rotation Above the
Region to be Rotated
• Rotate the same region
about y = 2
y 
x 2
y  e
x
• Notice that
y o u tsid e  rad iu s  2
r  2  y o u tsid e
Example 2 Line of Rotation Above the
Region to be Rotated
• The area of the larger
circle is
2
  2  y o u tsid e  
 2  e
x


2
V o lu m e   2  e
x

2
x
Example 2 Line of Rotation Above the
Region to be Rotated
• The sum of the volumes
is
n
V o lu m e 
  2
k 1
C

  2
 e
x

2
A
 1 6 .4 0 6 0 6 5
dx
 e
x

2
x
Example 2 Line of Rotation Above the
Region to be Rotated
• The area of the smaller
circle is
2
  2  y in sid e  

 2 
x 2

V o lu m e   2 

2
x  2

2
x
Example 2 Line of Rotation Above the
Region to be Rotated
• The sum of the volumes
is
n
V o lu m e 
  2 
x2
k 1
C

  2 
A
7 .8 7 0 3 6 0
x2

2
dx

2
Vx
Example 2 Line of Rotation Above the
Region to be Rotated
• The volume of the solid
is the difference
between the two
volumes
V o lu m e 
C

  2
 e

2
C
dx 
A
8 .5 3 5 o r 8 .5 3 6

A
2 
x  2

2
dx
Example 3 Line of Rotation to the Left
of the Region to be Rotated
• Line of Rotation: x = -3
• Use the same two
functions
• Create the reflection
• Draw the two disks and
mark the radius
Example 3 Line of Rotation to the Left
of the Region to be Rotated
• The radius will be an xdistance so we will have
to write the radius as a
function of y.
y 
y
2
x 2
y  e
 x 2
y  e
x  y
2
2
x
x
ln y  x
Example 3 Line of Rotation to the Left
of the Region to be Rotated
The radius of the larger
disk is 3 + the distance
from the y-axis
or
3 + (ln y)
Area of the larger circle is
  3  ln y 
2
Example 3 Line of Rotation to the Left
of the Region to be Rotated
• Volume of each disk:
V o lu m e    3  ln y 
V o lu m e 

D
B
2
y
2
  3  ln y  d y
Example 3 Line of Rotation to the Left
of the Region to be Rotated
• The radius of the
smaller disk is
• 3+ the distance from
the y-axis
or
3 + (y2 – 2)
• Area of the larger circle
is
 3  (y
2
 2)

2
Example 3 Line of Rotation to the Left
of the Region to be Rotated
• Volume of each disk:

V o lu m e   3  ( y
V o lu m e 

D
B
2
 2)
  3  (y
2

2
 2)
y

2
dy
Example 3 Line of Rotation to the Left
of the Region to be Rotated
• Difference in the
volume is

D
B
2
  3  (ln y )  d y 
 1 5 .5 3 8 o r 1 5 .5 3 9

D
B
 3  (y
2
 2)

2
dy
Example 4 Line of Rotation to the
Right of the Region to Be Rotated
• Line of Rotation:
x=1
• Create the
region, reflect the
region and draw
the disks and the
radius
Example 4 Line of Rotation to the
Right of the Region to Be Rotated
• Notice the larger radius
is 1 + the distance from
the y-axis to the
outside curve.
• The distance is from
the y-axis is negative so
the radius is
1  (y
2
 2)
Example 4 Line of Rotation to the
Right of the Region to Be Rotated
• Area of Larger
disk:
 1  ( y
2

• The volume of
the disk is

2
 2)
 1   y  2
2

2
x
Example 4 Line of Rotation to the
Right of the Region to Be Rotated
• Volume of all
the disks are
n
  1  ( y
2

2
 2) V y
k 1


D
B
 1  ( y
2
 2)

2
dy
Example 4 Line of Rotation to the
Right of the Region to Be Rotated
• Area of smaller
disk:
  1  (ln y ) 
2
• The volume of
the disk is
2
  1  ln y  V y
Example 4 Line of Rotation to the
Right of the Region to Be Rotated
• Volume of all
the disks are
n
  1  ln y 
2
Vy
k 1


D
B
2
  1  ln y  d y
Example 4 Line of Rotation to the
Right of the Region to Be Rotated
• Find the
difference in the
volumes

D
B
 1  ( y
2
 2)

2
dy 

D
B
2
 1  ln y  d y  1 2 .7 2 0 6 7