Volumes by Section
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Cross Sections of a Pyramid
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Objective: To
provide a toolbox of aids for teaching students about the volume by
section method. 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: A
particular platform is not needed.
However, there is a collection of animations which can be viewed in a
browser or other utility, such as Quicktime, for running animations and/or
movies.
Instructor's
Notes:
In basic geometry students learn the formulas
for the volume of a rectangular solid and the volume of a cylinder. In
preparation for determining volumes of more general solids these basic
formulas can be revisited to lay a foundation for determining the volume of
solids of revolution.
The term cross section can be used
to describe a typical portion of some object or body. In politics we often hear statements like 'a cross section of voters'
or pollsters will base election predictions on 'a cross section of a
population' that was interviewed. In the physical sciences, the term cross section
is used to describe a slice perpendicular to an axis of an object. In
this context a cross section can be used to represent a typical slice of
an object.
For a solid whose cross sections are the same shape
and size, the volume of the solid is computed as the product of the area
of the cross section times the height of the object. The height can be a
vertical distance or a horizontal distance as illustrated in Figures 1 and 2.
Note that the height is measured along the axis of the object
(perpendicular to the cross section).
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Figure 1. |
Figure 2. |
A general cylinder is a solid formed by translating a
cross section S along a line or axis that is perpendicular to S. Several
cylinders are shown in Figures 3a-d.
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Figure 3a.
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Figure 3b. |
Figure 3c.
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Figure 3d. |
In Figure 3a a metal angle iron is translated to form an
'edge protector' for a workbench. (Click here
for pictures of angle irons.) In Figure 3b an ellipse is translated
to form an 'elliptical can'. In Figure3c a triangle is translated to form a
prism. In Figure 3d, a slice of bread is translated to form a 'perfect loaf'.
See Figure 4 for a realistic loaf; ignore a few slices at each end to
visualize the corresponding general cylinder.
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Figure 4. |
The volume of a general cylinder is the area of the
cross section S that is translated times the distance through which the cross
section is translated. If A(S) is the area of the cross section of a general
cylinder which is translated a distance h, then the volume of the resulting
solid is given by
For more general solids in which the
cross section is the same shape, but not the same size, we use integral calculus
to determine the volume. Let A(x) represent the cross sectional area which we
assume varies continuously with x in [a, b]. We partition [a, b] into
subintervals with end points a = x0 < x1 < x2
< ... < xn = b. For each subinterval [xi,
xi+1] we arbitrarily choose a value ti. The volume of
the portion of the solid over interval [xi, xi+1] is
approximated by the volume of the cylinder with cross sectional area A(ti)
and thickness Dxi
= xi+1 - xi, that is by, A(ti)Dxi.
Hence the volume of the entire solid is approximated by the a sum of
volumes of cylinders given by
This approximating sum of volumes of cylinders is a Riemann
sum and hence as we take a limit with n, the number of subintervals
becoming arbitrarily large and the maximum length of a subinterval getting
arbitrarily small we have that the volume of the solid is
obtained by integrating A(x) from a to b:
The animation in Figure 5 shows some cross
sections of a solid whose base is in the xy-plane between the the x-axis and the
curve
over interval [0, 9].
Figure 5.
The cross sections are semicircles with radii
hence we have that the area of a cross
section is
It follows that the volume of this solid is
given by
Note: In the animation in Figure 5 we
really displayed only wire frame schematics of the shape of a typical
cross section. True cross sections have thickness. The pictures are to
illustrate how cross sections would be drawn, rather than showing actual
cross sections. This is a graphical convenience in order to generate the
animation in a reasonable way.
A small gallery
of demos for illustrating the generation of solids with a cross section of
fixed shape is available by clicking on SECTION-METHOD-GALLERY.
These animations can be used by instructors in a classroom setting or by
students to aid in acquiring a visualization background relating to the
generation of solids with cross section of a fixed shape. The demos provide a variety of animations for some common
examples.
Classroom Activities:
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An informative discussion about
using props as teaching aids is in Carol Critchlow's paper 'A prop
is worth ten thousand words', The Mathematics Teacher, Vol.
92, No. 1, January 1999. Several suggestions for motivating volumes of
solids with fixed shape cross sections are discussed.
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For a description of a a good
hands-on project involving volumes of solids with fixed shape cross sections
see Theresa Offerman's paper 'Foam Activities', The Mathematics
Teacher, Vol. 92, No. 5, May 1999. The discussion focuses on constructing
physical models for such surfaces using florist's foam which is light
weight, easily shaped, and inexpensive. Accompanying this paper is a set
of worksheets that can be used to provide guidance on how to construct a
model.
Credits:
This demo was constructed by Dr.
David R. Hill, Temple University for Demos with
Positive Impact.
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