Volumes by Section 

Cross Sections of a Pyramid

 
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).

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.

Figure 3a.

 

Figure 3b.

Figure 3c.

 

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.

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:

  • 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.

  • 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.


DRH 2/24/02     last updated 5/24/2006

Since 3/1/2002