1. Volumes of Solids of Revolution:
• Disk Method
• Washer Method

2. Make Sure You Remember Process for
Calculating Area
Divide the region into n pieces.
Approximate the area of each piece with a rectangle.
Add together the areas of the rectangles.
Take the limit as n goes to infinity.
The result gives a definite integral.

3. General Idea - Slicing
1. Divide the solid into n pieces (slices).
2. Approximate the volume of each slice.
3. Add together the volumes of the slices.
4. Take the limit as n goes to infinity.
5. The result gives a definite integral.

4. Disk Method

5. Volume of a Slice
Volume of a cylinder?
h
r
2
V r h


What if the ends are
not circles?
A
V Ah

What if the ends are not
perpendicular to the side?
No difference!
(note: h is the distance
between the ends)

6. Volume of a Solid
1
lim ( )
n
k
n
k
V A x x


 

a xk b
A(xk)
( )
slice k
V A x x
 
x

( )
b
a
A x dx
 
The hard part?
Finding A(x).

7. Volumes by Slicing: Example
Find the volume of the solid of revolution formed by rotating the
region bounded by the x-axis and the graph of from
x=0 to x=1, about the x-axis.
y = x

8. Here is a Problem for You:
Find the volume of the solid of revolution formed by rotating the
region bounded by the x-axis and the graph of y = x4, from x=1 to
x=2, about the x-axis.
Ready?
A(x) = p(x4)2= px8.

9. Washer Method

10. Setting up the Equation
Outer
Function
Inner
Function
R
r

11. Solids of Revolution
A solid obtained by revolving a region around a line.
When the axis of rotation is
NOT a border of the region.
Creates a “pipe” and the
slice will be a washer.
Find the volume of the solid
and subtract the volume of
the hole.
f(x)
g(x)
xk b
a
NOTE: Cross-section is
perpendicular to the
axis of rotation.
   
2 2
( ) ( )
b b
a a
V f x dx g x dx
 
 
 
   
2 2
( ) ( )
b
a
V f x g x dx

 


12. Example:
Find the volume of the solid formed by revolving the
region bounded by y = (x) and y = x² over the interval [0,
1] about the x – axis.
2 2
([ ( )] [ ( )] )
b
a
V f x g x dx

 

   
 

1
0
2
2
2
dx
x
x
V 
V = p (x - x4
)dx
0
1
ò
V = p
x2
2
-
x5
5
æ
è
ç
ö
ø
÷
0
1
=
3
10

13. Here is a Problem for You:
Ready?

14. So……how do you calculate
volumes of revolution?
• Graph your functions to create the region.
• Spin the region about the appropriate axis.
• Set up your integral.
• Integrate the function.
• Evaluate the integral.