The Disk Method for
Volumes of Solids of Revolution
|
|
Objective: To
provide a toolbox of aids for teaching students about the volume of solids by
the disk method. We include a variety of approaches which can be easily
adapted to different levels of instruction. In addition, a collection of
animations is included which can be run on a number of platforms.
Level: Calculus
courses in high school or college.
Prerequisites:
An introduction to Riemann sums and the use of integrals to determine the area
of a planar region. Basic ideas about volume. Platform: We
reference demonstration routines in Java, MATLAB, Mathematica, and Maple.
However, there is also a collection of animations which can be viewed
independent of a particular software platform.
Instructor's
Notes:
For this demo we consider a region R bounded by a
(continuous) curve y = f(x) over an interval [a, b], the x-axis, and possibly
one or two vertical lines x = a and x = b. We further require that f(x) be
nonnegative over [a, b]. Figure 1 displays several regions of this type.
Figure 1.
The region R will be revolved about the x-axis
to obtain a solid of revolution. A major objective is to provide visualization
demonstrations to enable students to more clearly understand the procedure for
the computation of the volume of the resulting solid by the disk method.
Using Props ('A prop is
worth ten thousand words', see [x].)
A variety of props, objects you can handle, are quite
useful for helping students visualize surfaces of revolution and the components
of the disk method.
A sheet of paper and a length of dowel. Take a
sheet of paper, fold it in half. Now cut a shape of a curve along the open
end through both layers. Staple the folded sheet in several places along the
curve you cut. Insert a length of dowel wood along the fold to act as an axis of
revolution. See Figure 2.
Figure 2.
Now you are set to have your students see you create a
solid of revolution. (Of course they will need to use their imagination a
bit.) In class you point out the planar region which is the paper cut out.
Next rotate the region about the dowel (or better yet have a student do it).
Ask for a description of the solid generated by rotating the region. You may
want to lead them to indicate a vase shaped object. A vase created on a
potter's wheel is a nice example of solid of revolution generated by
rotation about a vertical axis.
Make a vase.
Figure 3.
Wedding bell.
The common paper wedding bell (see Figure
4) nicely illustrates a solid of revolution. It is easily transported to the
classroom and can be used both by the instructor and the student.
Figure 4.
Figures 5 - 7 illustrate the generation
of a solid of revolution about a horizontal axis. Here we held the bell
against a vertical board and let it open by itself. The paper folds
uncompress illustrating the rotation of the planar region shown in Figure 5
into the 'solid bell'.
Figure 5. |
Figure 6. |
Figure 7. |
Fresh fruits and vegetables.
The produce section of your favorite
market is a good source of solids of revolution. Cucumbers and squash
illustrate solids of revolution of the type discussed in this demo. For
simple illustrations we often draw curves like those in Figures 8 and 9 on a
set of axes and ask students to imagine the surface generated as we revolve
the region R about the horizontal axis.
Figure 8. |
Figure 9. |
The solid generated by revolution by the
region R in Figure 8 looks like a cucumber; see Figure 10. The solid
produced from the region R in Figure 9 looks like a squash; see Figure
11.
Figure
10. |
Figure 11. |
Other produce like oranges and melons
also provide nice models for solids of revolution. See Figures 12 and 13.
Figure 12.
|
Figure 13.
|
The Disk Method.
For the regions R in this demo we
"slice" the planar region into strips of area and revolve the strips
around the x-axis to generate "disks". The disks are cylinders which
approximate the volume of the actual slices of the solid generated by the
revolution of region R. Figure 14 shows a curve and then one strip of area
which generates the cylindrical disk displayed in Figure 15. The circular
faces of the cylindrical disk are called the cross sections of the
solid.
Figure 14.
Figure 15.
Using our cucumber prop shown in Figure 10
a collection of disks is shown in Figure 16. The actual volume of a slice of
cucumber is approximated by the volume of a cylindrical disk as in Figure 15.
The volume of the disk in
Figure 16.
Figure 15 is computed as the area of the
circular face of the disk times it thickness. The area of the circular face is
p times (radius)2
. In Figure 17 we show how this relates to the curve from Figure14. We see that the radius is a value of the function f(x) describing the curve and the
thickness is an increment along the x-axis.
Figure 17.
In the general case we point out that the solid is sliced
into approximating cylindrical disks. The kth disk has radius rk and thickness
Dxk so for a solid generated by revolving about the x-axis,
the volume of the kth approximating disk is
 .
Summing the volumes of the disks as we
require the thickness to get smaller and smaller leads to the standard
integral formula for the disk method:
 .
We illustrate the steps of the process
described above with the following animation which generates a solid like the
wedding bell in Figure 7.
A small gallery
of demos for illustrating the disk method for volumes of solids of revolution
is available by clicking on DISK-METHOD-GALLERY.
These animations can be used by instructors in a class room setting or by
students to aid in acquiring a visualization background relating to the steps
of disk method. The demos provide a variety of animations for some common
examples. Also included is a step-by-step narrative script of the displays in the
animations with pictures that illustrate a step.
Classroom Activities:
http://www.jamesrahn.com/CalculusI/Activities/disk_method.htm
Here you will see Jim's approach
to the disk method using apples and oranges. You will also see his class
having "tasteful fun" finding
volumes by the disk method.
-
An informative discussion about
using props as teaching aids is in Carol Critchlow's paper 'A prop
is worth ten thousand words', Mathematics Teacher, Vol.
92, No. 1, January 1999.
-
For a description of a good
hands-on project involving volumes of revolution see Judith Schimmel's paper
'A New Spin on Volumes of Solids of Revolution', Mathematics Teacher,
Vol. 90, No.9, December 1997. This work incorporates modeling and employs
both a calculator and computer software.
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 disk
method.
-
A MATLAB routine, diskmethod.m,
for illustrating the basic steps of the disk method is available for
download by clicking here. To see a
description of the routine click here.
Animations used in this demo were generated with diskmethod.m by capturing
the evolving pictures and then editing the video files into various formats.
The routine was written by David R. Hill and Lila F. Roberts.
The development in this project
leads students to compute the volume of a cup (or vase) as illustrated in
Figure 3 above.
Credits:
This demo was constructed by David R. Hill for Demos with Positive Impact.
Dr.
David R. Hill
Department of Mathematics
Temple University
|
