Objective:  The purpose of this demo is to illustrate the use of the method of cross sections to find estimates of the volume of physical objects.  The objects in this demo are not solids of revolution.

Level:  This demo is appropriate for calculus and numerical analysis courses.

Prerequisites:  Students should have knowledge of the method of cross sections for finding volume.  In the DEMOs with POSITIVE IMPACT project at URL 

http://mathdemos.gcsu.edu/solids/ 

you will find a collection of demonstrations of basic calculus techniques for computing volumes of solids.

Although the demo involves interpolation to a set of data points, a calculus student need not have prior knowledge of this technique for curve fitting since this aspect of the demo is automated and hence transparent to the student.  In a numerical analysis course this demo can be presented as an application of interpolation as well as the use of calculus. 

Platform:  The demo presented here utilizes Mathematica.   Mathematica 5 notebooks are provided.  The notebooks can be modified to approximate volumes of other objects for which a cross section approach is appropriate and a digital photograph is available.  

Instructor's Notes:  This demo is designed to illustrate "practical" uses of the cross section method for computing volumes of solids.  We present two techniques, one using a sum of approximating volumes of cross sections and another using integration of approximating functions.  These techniques are applied to two physical objects:  an unusually shaped wine bottle and a sea worm.

DEMO 1:

The idea:  We want to approximate the volume of the wine bottle shown in Figure 1. This wine bottle was chosen because it has an interesting shape that cannot be considered as a solid of revolution.  (This brand of wine is widely available for purchase.)  We recommend bringing the (empty) bottle to class so that students have a chance to see and hold it.

Slicing the bottle parallel to its bottom, we see that the cross sections are circular.  Thus, we can approximate the volume using a section method. 

Figure 2. Wine Bottle and Circular Cut

The approximation procedure involves first selecting points along the upper edge of the photograph and then generating a function that goes through each of the points.  call this function y = top(x).  This function is called an interpolating function.  We repeat the process along the bottom edge and call this interpolation funcxtion y = bottom(x).  The animations in Figure 3 illustrate the process.

Figure 3.  Select points along the top and bottom; Fit with interpolating functions.

The two interpolation functions are shown in Figure 4.  NOTE:  In a numerical analysis class, it is appropriate to discuss the choice of interpolating function in detail.

Figure 5 shows a three-dimensional model of the bottle.  Click on the figure to turn.

 

Figure 5.  A Mathematica generated model of the wine bottle.

Now we partition the domain of the bottle model as x₀ < x₁ < x₂ < ... < x_n, where x₀ is at the left end of the bottle and x_n at the right end.  Next we compute one half the distance between top(x_i) and bottom(x_i), where x_i is the left end point of a subinterval of the partition.  This gives us the radius of the approximating disk whose thickness is .  This is illustrated in Figure 6.

 

Figure 6.  Approximation Sequence.

An approximation of the volume can be obtained by summing the volumes of the disks.  We do need to be concerned about units; our data was obtained by selecting pixels from a jpeg photograph.  This photograph has a resolution of 72 pixels per inch so our measurements need to be scaled.  An approximation using 20 disks is computed as follows:

Note that 60.2295 in² is about 988 ml. (1 in² is approximately 16.387 ml)

We can also use an integral formulation with the interpolation functions to approximate the volume:

,  about 970.96 ml.  In this formula, r(x) = top(x) - bottom(x).

An important part of the discussion involves sources for error--it would be helpful to actually measure the volume of the bottle by filling it with water and then pouring it into a graduated cylinder.  In a numerical analysis class it is appropriate to spend more time on the error discussion and to distinguish between errors due to the measurement and errors due to the approximation techniques.

DEMO 2:  SEA WORM  (CAN ALSO BE USED AS A PROJECT IF MATHEMATICA IS AVAILABLE TO STUDENTS)

Worm mass is measured as ash-free dry weight.  Volume is more strongly correlated with mass than other measurements.  However, when doing growth experiments, you can't measure ash-free dry weight initially (worms don't grow after being turned to ash at 500 degrees C!).  

Using the ideas generated in the previous demo, we can now focus our attention on approximating the volume of the sea worm.

We use the same approach as in the wine bottle example to select points along the outline of the worm (it is tricker here to get nice selection of points). 

The three-dimensional depiction and an animation showing the cross section approximation are shown in Figure 9.

Figure 9.  Worm Volume Approximation

Once again, we can use either obtain an approximation of the volume by summing the volumes of the approximating cross sections or use integrating the interpolation functions directly.

Mathematica Notebooks can be viewed and downloaded from here.

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CREDITS

This demo was developed by

Lila F. Roberts
Georgia College & State University

but it was inspired by Judy O'Neal's demo Enlightening Volumes:  Curve Fitting to Approximate Volumes and the wine bottle given to Lila by a bartender in Phoenix, AZ.

Many thanks to Eric Weissberger at University of Maine Darling Marine Center for his query about volumes of sea worms and for his generous contribution of the digital photographs of two stunning sea worms.

More thanks to Jim Braselton at Georgia Southern University for his Mathematica expertise and willingness to share!

LFR 12/31/2003;   lastupdated 1/06/2004 DRH

Visitors since 1/06/04