7.2 Volume: The Disc and Washer Method
We have formulas to determine the volume of many regular solids such as cylinders, cones, rectangular solids,
and spheres, but how do we determine the volume of a solid that does not fall into one of these standard
categories? Consider the solid created by revolving a region in a plane about a line. The solid generated is
called a solid of revolution.
(courtesy of International Educational Software)
Consider revolving the curve shown above around the x-axis.
Follow the hyperlink http://www.ies.co.jp/math/products/calc/applets/rotate/rotate.html to view an animation of a curve
revolving around an axis and a demonstration of using a series of discs (cross sections) to determine the
volume.
a
xi
b
A cross section perpendicular to the axis of revolution would form a disc. We could determine the volume of
the disc with the formula:
Volume of the disc =
(area of the disc)(width of the disc)
!R 2 w
To determine the volume of the solid, consider dividing the solid into a series of such discs where the radius
would be equal to the value of the function for the particular x value for a given disc. We could add the
volumes of the discs and approximate the volume as:
Volume of the solid
!
n
! # [ R( x )] "x
2
i
i =1
We could more closely approximate the volume of the solid by increasing the number of discs, and in fact,
allowing the number of discs to go to infinity:
n
Volume of solid
Scherer
=
Page 1
lim ! % [ R( xi )] $x
n #"
i =1
2
b
=
" ! [ R( x)]2 dx
a
2/15/2010
Similarly, we can determine the volume of a solid generated by revolving a region in a plane about the y-axis.
In this case, the representative rectangle would be drawn perpendicular to the y-axis and you would integrate
with respect to y.
d
n
Volume of solid
=
lim ! % [ R( yi )]2 $y
n #"
=
i =1
" ! [ R( y )]2 dy
c
Washer Method
Observe that the region in the plane that is revolved about the x-axis in the example above is adjacent to the xaxis. Consider a region that is located some distance from the x-axis. The volume of a solid of revolution that
is generated by revolving the figure about the x-axis in this case can be approximated by the sum of the volumes
of the "washers" that are created. Consider finding the volume of a representative washer by determining the
volume using the outer radius and subtracting the volume determined by using the inner radius. In effect, you
subtract the "hole" from the entire disc.
a
b
Volume of Washer
=
(volume of the entire disc) - (volume of the hole)
Volume of Washer
=
!R 2 w " !r 2 w
Volume of Solid
=
# ! ( [ R( x)]2 " [r ( x)]2 )dx
b
a
The washer method also applies to the case where a region in a plane is revolved about the y-axis.
d
Volume of Solid
=
# ! ( [ R( y )]2 " [r ( y )]2 )dy
c
Scherer
Page 2
2/15/2010
Finally, we need to consider the case when we revolve a region in a plane about a line other than the x- or yaxis. Depending on the orientation of the region and the axis of revolution, it may be appropriate to use either
the disc or the washer method. The graphs below demonstrate both cases.
Disc method with axis of revolution other than x- or y-axis
f(x)
g(x)
a
b
Now the radius is defined as:
R(x) = f(x) – g(x)
b
Volume = " ! [ R( x)]2 dx
And the volume is determined by:
a
Washer method with axis of revolution other than x- or y-axis
f(x)
g(x)
c
a
b
Now the outer radius is defined as:
R(x) = f(x) – c
And the inner radius is defined as:
r(x) = g(x) – c
And the volume is:
Volume = # ! ([ R( x)]2 " [r ( x)]2 )dx
b
a
NOTE: When the axis of revolution is “above” the function, then R(x) = c – g(x) and r(x) = c – f(x). When the
axis of revolution is vertical, in general, you use x as a function of y and integrate with respect to y. The moral
to the story here is, don’t simply relay on formulas, be sure that you understand the geometry of each problem
and set up your integral based on the geometry of the graph.
Scherer
Page 3
2/15/2010
Example 1
Disc Method
Find the volume of the solid of revolution generated when you revolve f(x) about the x-axis, between 0 and !.
f ( x ) = sin x
Example 2 Disc Method
Find the volume of the solid of revolution generated when you revolve the area bounded by f(y) and x = 0
between y = 0 and y = 1 about the y-axis.
f ( y) = x
Scherer
2
3
Page 4
2/15/2010
Example 3 Washer Method
Find the volume of the solid of revolution generated when you revolve the area between the two functions f(x)
and g(x) about the x-axis.
f ( x) = ! x 2 ! 2 x + 3
g( x) = x + 3
Example 4 Washer Method / Axis of revolution other than x- or y-axis
Find the volume of the solid generated by revolving the area bounded by y = 1/x, y = 0, x = 1, and x = 4 about
the line y = 4.
Scherer
Page 5
2/15/2010