**Objective**: The purpose of this demo
collection is to help students to understand the concepts that motivate the
elements of computation of volumes of solids of revolution. Rather than
introduce volumes of solids of revolution by a purely formula-driven approach,
these demos provide the opportunity for visualization of the basic approximating
elements that lead to the standard calculus expressions that, when computed,
give the desired volumes. The demos in this collection were developed to
provide a toolbox of aids that instructors have found to be effective for
teaching students about volumes of solids. We include a variety of approaches
which can be easily adapted to different levels. In addition, a collection of
animations are included that can be run on a number of platforms.
**Level: **
These demos can be presented in any course in which volumes of solids
are introduced.
**Prerequisites:
**Students should be familiar with basic planar area computations such
as areas of circles, triangles, and rectangles. In addition, basic
volume computations such as volumes of cylinders, volumes of rectangular
parallelepipeds, etc. are useful. Students should also be familiar
with areas of planar regions using approximations by Riemann sums and
limits that lead to the definite integral.
**Platform: **
Interactive MATLAB routines and *Mathematica* notebooks that
illustrate volumes modeled using disks and shells are given. A
gallery of animations that run in web browsers is provided. In
addition, several physical props are suggested.
**Instructor's Notes:** We
tell our calculus students that the "second" major topic is the
"area problem". In examples and demos we stuff rectangles or
trapezoids in or around a region and argue that by taking a special type of
limit we obtain an integral expression that represents the area of the region.
The demos below illustrate the development of areas of planar regions by
approximations that lead to Riemann sums (and the definite integral
representations of the areas).
A natural extension
of the area of a planar region is the volume of a solid. We usually rely
on a students visualization skills as we banter about terms like
"cross sections of equal area", or "revolve the region
about an axis to obtain a solid of revolution". In each of these
cases we wave our hands, make (often) crude sketches, and claim that we
dissect the solid in such a way that we can sum up the volumes of each
piece to obtain an approximation of the volume of the whole solid. We
tell students that we can cut the solid into pieces that have a known
cross section or, in the case of solids of revolution, we dissect the
surface of revolution into "disks", "washers," or
"cylindrical shells".
The difficulty with
our usual classroom approach is that the visualization skills of many
students are not well-developed when it comes to three dimensional
objects. To provide better opportunities for a student to improve such
skills we present a collection of demos that provide a variety of
instructional aids. These include physical objects, computer generated
sketches, and computer generated animations. Combinations of such aids
can provide students with an opportunity to sharpen visualization skills
and then to make informed decisions on how to proceed with the
calculations for volumes of solids encountered in an introductory
calculus course.
**
Volumes
by Section**
deals with solids that can be sliced into pieces with a known cross
section. While this broad class of solids includes solids of
revolution that can be sliced into disks or washers, in this demo we
focus on solids that are not necessarily solids of revolution. The
k^{th} approximating element has volume computed by multiplying
the area of the cross section, A_{k}, by the thickness
so the volume of the approximating element is
.
**
The
Disk Method for Volumes of Solids of Revolution**
concerns solids generated when a planar region is revolved about the
x-axis. In particular, if a region bounded a curve y = f(x) and the
domain interval [a,b] is revolved about the x-axis, the resulting solid
may be sliced into disks. The k^{th} disk has radius r_{k}
and thickness so for a
solid generated by revolving about the x-axis, the volume of the k^{th}
approximating element is
.
**Solids
of Revolution: The Method of Shells** involves revolving a region bounded by a curve y = f(x) and the domain
interval [a,b] about the y-axis. The solid is then filled with
cylindrical shells. The k^{th} approximating element is a
cylindrical shell that has average radius r_{k}, height h_{k},
and thickness .
Thus, for a
solid generated by revolving about the y-axis, the k^{th }approximating
element has volume
.
This demo also
includes method of shells visualizations for the solid of revolution
formed by revolving a region bounded by two curves about the y-axis.
**
The Washer Method for
Volumes of Solids of Revolution** involves revolving a region revolved about
one of the coordinate axes. In this demo, the
resulting solid has a "hole." The k^{th}
approximating element is a washer that has inside radius r_{in},
outside radius r_{out}, and thickness
.
Thus for a solid generated by revolving about a coordinate axis, the k^{th}
approximating element has volume
.
The symbolic form of
the volume element depends on whether the axis of revolution is the x-axis
or y-axis. Visualizations for revolution about both x and y axes are
provided.
A
gallery
of animations has been developed to accompany the demos. In
addition to animated gifs that run in a browser, movies (mov format) are
included.
__References__
1. Carol M. Critchlow,
"A Prop is Worth Ten Thousand Words," *Mathematics Teacher*,
92(1), Jan. 1999, pp 27-29.
2. Theresa Reardon Offerman,
"Foam Images," **Mathematics Teacher**, 92 (5), May
1999, pp 391-399.
3. James Rahn, "Giving Meaning to
Volume in Calculus," **Mathematics Teacher**, 84 (2), Feb.
1991, pp 110-112.
4. Judith Schimmel, "A New Spin on
Volumes of Solids of Revolution," *Mathematics Teacher*,
90 (9), Dec. 1997, pp 715-717.
**Credits**: This demo
collection was organized by Dr. David R. Hill and Dr. Lila F.
Roberts. MATLAB files to accompany the demos were written by David
Hill; *Mathematica *notebooks were written by Lila Roberts.
David
R. Hill
Mathematics Department
Temple University
Philadelphia, PA
Lila F. Roberts
College of Information &
Mathematical Sciences
Clayton State University
Morrow, GA 30260
The gallery of animations were generated by Drs. Hill and Roberts. |