How To Create A Solid Of Revolution Animation

In a contempo blog post , I pointed out that Maple did non take a built-in functionality for drawing graphs that arise in computing volumes past slices. However, I did provide several examples of advertisement-hoc visualizations that one could build with the graphing tools in Maple.

Recently, a user called attending to a weakness in the Student Calculus 1 control, VolumeOfRevolution . This command (and the tutor built on it) will draw a surface of revolution divisional past the surfaces generated by revolving the graph of 1 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 most the line .

The VolumeOfRevolution  command generates the graph in Effigy 1.

Figure ane    Figure generated past VolumeOfRevolution  command

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

(1)

from the definite integral

(2)

is indeed correct. Unfortunately, Figure 1 does non prove the tertiary bounding surface generated when  along the -centrality 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 ended that there was something incorrect with the command.

It helps to know that the command was written before the "transparency" option was bachelor in Maple. Hence, the coding was deliberate - the tertiary bounding surface was not included because it was thought that it would non be possible to see "within" the solid. Apparently, this shortcoming is now recognized, and a gear up for this command may well be forthcoming in the future.  Meanwhile, Effigy ii shows what could be done by combining Effigy i with an appropriate bounding cylinder.

Figure 2    Outer bounding-cylinder added to Figure 1

An equivalent example is the following.

Summate the volume of the solid of revolution generated when the region bounded past the curves  and the -centrality is rotated about the line .

Effigy three is the graph generated by the VolumeOfRevolution  command with the two input functions  and 0. (If the second curve, 0, is non included, neither the figure nor the computed book will be correct.)

Effigy 3    Figure generated by VolumeOfRevolution  command

The red surface is generated past the rotation of , well-nigh the line ; the green surface, by . This time, the missing 3rd surface is the cylinder of radius i, internal to the 2 surfaces visible in Figure three. It'southward the same issue faced in Figure ane, but hither, the impact of the missing boundary is not every bit severe.

But the affair I really wanted to bespeak out in this communciation is an insight that arose from my musings on these examples. The restriction of the VolumeOfRevolution  command to just 2 functions is mitigated past making at to the lowest degree ane of these ii functions a piecewise function. Thus, the command tin be made to draw the "solid" of revolution in the following instance.

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 outset glance, information technology seems as if there are besides many bounding curves for this region to be handled by the VolumeOfRevolution  command. However, if the piecewise function

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

Effigy 4    Graph of the piecewise part

Taking this piecewise function equally  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 however missing, but the utilise of the piecewise function certainly extends the applicability of the VolumeOfRevolution  command.

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

Sometimes, equally in Figure six, you can observe a way to obtain a improve approximation of what you really desire. This figure is a graph of  fatigued in cylindrical coordinates with the "filled" option used to shut the surface downward to the  aeroplane.

Effigy 6    The solid in Figure 5 fatigued with all bounding faces

It is then non a big spring to the animation in Figure 7.

Figure 7    Animated cartoon of the solid shown in Figure 6

  Annotation: The one yellow cell and  all the plots contain subconscious input. To meet the hidden input, open up the Table Properties dialog and uncheck the "hide contents" checkbox. I notice this device to exist ane 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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Source: https://www.mapleprimes.com/maplesoftblog/90006-Drawing-Solids-Of-Revolution

Posted by: smithbutch1974.blogspot.com

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