## Drawing Solids of Revolution

Maple

In a recent blog post, I pointed out that Maple did not have a built-in functionality for drawing graphs that arise in computing volumes by slices. However, I did provide several examples of ad-hoc visualizations that one could build with the graphing tools in Maple.

Recently, a user called attention to a weakness in the Student Calculus 1 command, VolumeOfRevolution. This command (and the tutor built on it) will draw a surface of revolution bounded by the surfaces generated by revolving the graph of one or two functions.

The user provided an example similar to the following.

 Find the volume of the solid of revolution generated when the region bounded by the curves , and the -axis is rotated about the line .

The VolumeOfRevolution command generates the graph in Figure 1.

Figure 1   Figure generated by VolumeOfRevolution command

The first and second functions given to this command generate surfaces colored red and green, respectively. The volume that would be computed by the command

 (1)

from the definite integral

 (2)

is indeed correct. Unfortunately, Figure 1 does not show the third bounding surface generated when  along the -axis is rotated about . Hence, it appears as if the "solid" is where nothing is graphed, and where there are surfaces drawn, no integration takes place. Hence, the user concluded that there was something wrong with the command.

It helps to know that the command was written before the "transparency" option was available in Maple. Hence, the coding was deliberate - the third bounding surface was not included because it was thought that it would not be possible to see "inside" the solid. Apparently, this shortcoming is now recognized, and a fix for this command may well be forthcoming in the future.  Meanwhile, Figure 2 shows what could be done by combining Figure 1 with an appropriate bounding cylinder.

Figure 2   Outer bounding-cylinder added to Figure 1

An equivalent example is the following.

 Calculate the volume of the solid of revolution generated when the region bounded by the curves  and the -axis is rotated about the line .

Figure 3 is the graph generated by the VolumeOfRevolution command with the two input functions  and 0. (If the second curve, 0, is not included, neither the figure nor the computed volume will be correct.)

Figure 3   Figure generated by VolumeOfRevolution command

The red surface is generated by the rotation of , about the line ; the green surface, by . This time, the missing third surface is the cylinder of radius 1, internal to the two surfaces visible in Figure 3. It's the same issue faced in Figure 1, but here, the impact of the missing boundary is not as severe.

But the thing I really wanted to point out in this communciation is an insight that arose from my musings on these examples. The restriction of the VolumeOfRevolution command to just two functions is mitigated by making at least one of these two functions a piecewise function. Thus, the command can be made to draw the "solid" of revolution in the following example.

 Calculate the volume of the solid of revolution formed when the region bounded by the graphs of , and the -axis is rotated about the line .

At first glance, it seems as if there are too many bounding curves for this region to be handled by the VolumeOfRevolution command. However, if the piecewise function

is defined, we see that its graph, shown in Figure 4, describes the region in question.

Figure 4   Graph of the piecewise function

Taking this piecewise function as  and , the VolumeOfRevolution command generates "solid" shown in Figure 5. Of course, the "central" cylinder corresponding to the rotation of the line segment along  is still missing, but the use of the piecewise function certainly extends the applicability of the VolumeOfRevolution command.

Figure 5   Complex surface of revolution generated by using a piecewise-defined function in the VolumeOfRevolution command

Sometimes, as in Figure 6, you can find a way to obtain a better approximation of what you really want. This figure is a graph of  drawn in cylindrical coordinates with the "filled" option used to close the surface down to the  plane.

Figure 6   The solid in Figure 5 drawn with all bounding faces

It is then not a big leap to the animation in Figure 7.

Figure 7   Animated drawing of the solid shown in Figure 6

Note: The one yellow cell and  all the plots contain hidden input. To see the hidden input, open the Table Properties dialog and uncheck the "hide contents" checkbox. I find this device to be one of the better ways to display computed math without having to display the ccorresponding input, yet preserving that input for the interested reader.

RJL Maplesoft

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