| For this demo we focus on
regions revolved about the x-axis or y-axis such that the
resulting solid of revolution has a "hole," as illustrated in
Figure 1.
|
Region to
Revolve about x-axis |
Resulting Solid |
|
Region to Revolve about y-axis
|
Resulting Solid |
Figure 1. Solids of
Revolution |
If we slice the solid
perpendicular to the axis of revolution as in Figure 2, the
resulting solid resembles a washer, hence the technique for
calculating the volume of the solid of revolution is called the
washer method.
|
 |
 |
Figure 2.
The washer method is based upon
approximating the volume of the solid by adding the volume of
the individual washers. The approximation process involves
generating a partition and constructing the washers; the
animation in Figure 3 shows the generation of a partition and
construction of one of the approximating washers.
Figure 3.
Some Useful Props
To motivate the ideas central
to the washer method, there are several props in addition to
real washers that can be useful as visualization tools.
LIFESAVERS candies (Figure 4)
are easy to bring to class and share with students as the
discussion begins. Before indulging, it is easy for students to
hold the candy and think about how one could compute the volume
of the candy by imagining that the candy is a disk and then
subtracting the "hole."
Figure 4.
Some tasteful props include
angel food or bundt cake (any cake baked in a tube pan). By
slicing the cake perpendicular to the hole, nice examples of
washers (Figure 5) can be eaten later.
Figure 5.
Another use of the angel cake
model is to illustrate differences in using washers to
approximate volumes and using shells. Figure 6 displays a cake
washer and a cake shell. Note that the washer is sliced
perpendicular to the hole (axis of revolution) while the
orientation of the shell is parallel to the hole (axis of
revolution).
 |
 |
Figure 6.
CD-R disks often are sold in
spindles of 50 or 100. The CDs on a spindle, shown in Figure
7, provide an easy to use way to illustrate how the washers
stack up to form a larger solid.
|
Figure 7.
Approximating the Volume
As in the disk method, the washer
method approximates the volume of the solid by adding the volumes of
typical slices. We begin by determining the volume of a typical
slice.
Figure 8 illustrates how a washer
can be generated from a disk. We begin with a disk with radius rout
and thickness h. A smaller concentric disk with radius rin
is removed from the original disk. The resulting solid is a washer.
Figure 8.
The volume of the washer is
calculated by subtracting the volume of the inner disk from the volume
of the larger disk. Thus,
.
For a solid of revolution generated
by revolving a region about the x-axis, the inner and outer radii are
functions of x. Consider a region for which the upper boundary is the
graph of y = f(x) and lower boundary is the graph of y = g(x) for x =
a to x = b (Figure 9).
Figure 9.
The region is partitioned into n
parts with width
,
where i = 1, 2, ..., n. Let xi be a value in the ith
interval [xi-1,xi]. The outer radius is given
by rout = f(xi) and the inner radius is rin
= g(xi) (Figure 10).
The ith washer and its
volume are shown in Figure 11.
Figure 11.
By allowing the thickness of each
washer to become very small and summing up the volumes, we obtain the
definite integral representation for the washer method:
.
For a solid of revolution generated
by revolving a region about the y-axis, the inner and outer radii are
functions of y. Consider a region for which the right boundary is the
graph of x = F(y) and left boundary is the graph of x = G(y) for y = c
to y = d (Figure 12).
Figure 12.
The region is partitioned into n
parts with thickness
, where i = 1, 2,
..., n. Let yi be a value in the ith interval
[yi-1,yi]. The outer radius is given by rout
= F(yi) and the inner radius is rin = G(yi)
(Figure 13).
Figure 13.
The ith washer and its volume are
shown in Figure 14.
Figure 14.
By allowing the thickness of each
washer to become very small and summing up the volumes, we obtain the
definite integral representation for the washer method:
.
When we developed the
disk method for computing volumes we illustrated the generation of
the disks by revolving a rectangle about an axis. In a similar way, a
washer is generated by revolving a rectangle about the axis of
revolution. This is shown in the animations in Figure 15.
Figure 15.
The animation in Figure 16
illustrates the sequence of steps involved with the washer method for
computing the volume of the solid of revolution generated by revolving
a region in the first quadrant bounded by the graphs of y = 1, x = 0,
and y = x2 about the x-axis. First, the region is
partitioned and a typical washer is drawn. Approximating half-washers
are drawn. To complete the visualization, the approximating washers
are produced. Finally, the solid of revolution is generated.
Figure 16.
The animation in Figure 17
illustrates the sequence of steps involved with the washer method for
computing the volume of the solid of revolution generated by revolving
a region in the first quadrant bounded by the graphs of y = x2
and 
about the
y-axis First, the region is partitioned and a typical washer is
drawn. Approximating half-washers are drawn. To complete the
visualization, the approximating washers are produced. Finally, the
solid of revolution is generated.
Figure 17.
Technology Resources:
There are a variety of
resources that employ calculators or software for illustrating and
computing volumes of solids of revolution. Following is a sample of
such resources which can be located using a search engine. We have
chosen ones that relate to the washer method.
