Is there a place to buy physical models to demonstrate the Calculus shell, disk, and washer methods?

I know a math teacher who is going to teach a calculus class that will include the shell, disk, and washer methods for calculating volumes. My question is, is there some 3D kit she could use to demonstrate these methods physically?

Ideally for example, the washer method could be demonstrated with a cylinder that has an inner cylinder that could be removed.

• This isn't interesting enough to warrant answer status, but I really like using toddler/baby stacking cups because they're very durable. Usually they are cylinders but this one is interestingly not. – kcrisman Jan 14 '19 at 14:21

6 Answers

3D printing is an attractive avenue. I think there is something to be gained by actual physical models.

Image from Elizabeth Denne's webpages.
See also Rebecka Peterson's Epsilon-Delta blog for low-tech alternatives:

Student volume models based on cross-sections.

It is probably easier to make 3D computer images of such methods, using even something simple like geogebra (or using CAD software if you know how to use it), that can be projected during class, than to obtain functional physical models. The latter can be machined or made using a 3D printer, but my impression is that computer visualizations are more adaptable and adjustable, and it's not clear to me that much is gained by using an actual physical model. In most any university physics or engineering department there is someone who knows how to make such physical models, but probably it is a better use of one's time to learn to use software designed for creating visualizations. (It is also cheaper - 3D printers and 3D printing is expensive - there is free software that can be used). Moreover, computer models can be interactive in ways that physical models simply cannot. The student can see a surface of revolution as it is generated, and so forth, and can tune parameters. Physical models will be made for a (usually fairly small) finite set of surfaces of revolution; with a computer model one can visualize the surface generated by any admissible curve.

On the other hand, some students probably benefit from touching an object. The intersection of two cylinders is good example of a surface for which a computer vizualization is probably less communicative than a physical model, simply because the surface is somehow quite hard to see.

One recommendation would be to look into using something like geogebra. The sorts of models you describe (appropriate for a multivariable calculus class) are not hard to realize in such software and there are already many examples available online. Making quality, usable physical models requires skills and access to facilities not all of us have, and in this sense computer visualizations are more easily prepared/generated.

• This has led me to wonder if research has been done on the advantages and disadvantages of physical models over 3D computer models. A quick search turned up a paper in the context of learning imaging anatomy, but what about the mathematics classroom? – J W Dec 24 '18 at 14:59
• @JW: It seems to me an interesting question, but I don't know any relevant literature. However, if someone like Joseph O'Rourke thinks that something is gained by using physical models (as his answer suggests he does), I take that seriously, because, judging from all that he has written and taught, he has lots of relevant experience and good judgment. – Dan Fox Dec 24 '18 at 17:15

Another substitute available nowadays: computer-animated videos. They should be just as convincing as physical models.

Search "shell integration animated" on YouTube to start. You may have to search many of the results to find one that you want to use.

I suppose, if you wish to illustrate the washer method then perhaps slicing a pear (see here for a picture into 10 or so horizontal slices would give you bunch of near washers with which you could estimate the volume of a pear. Then, you could even estimate the error in the calculation by the water displaced by the uncut pear. Ideally, you'd like a near circular pear, but you could estimate a mean radius if it was lumpy.

For cylindrical shells, I have a less fruity idea involving foam sheets held together by large rubber bands around some center cylinder. But, I haven't had a chance to try it out myself yet. Onions are almost good, but the layers are too curved. If you could find some cylindrical onions. See, someone needs to grow cylindrical onions. Perhaps there is some other easily obtained food which illustrates the shell method.

EDIT: Actually, the foam sheets I had in mind were way to slim to make this plausible. Perhaps Bologna would work for the right shape. It has a regular thickness and it easy to cut. I'm not sure how to arrange it to stay together.

Links break. As a mod on another Stack, and a blogger, I'm painfully aware of this. Please consider this a complement to Gerald's answer. i.e. He suggested a search instead of a hard link.

This particular very short video Integration and Volume - Shell Method Animation was ok, as a nice example. But, as it was created with Desmos, I found I was able to 'steal' the graph equations to my own account. The Desmos link is Shells.

An example of the graph

The most inexpensive and engaging material that you can easily slice is cake.

If you purchase a Bundt cake pan, you can easily create something slice-able that is (roughly) a solid of revolution.

• When you slice it in one direction (parallel to the table), you get washers. They are fairly compelling washers, too. You will be able to hold them up.
• When you slice it another way (awkwardly around the outside), you get cylindrical shells -- and, if your cake is moist enough, you will be able to unroll the shell to see that it roughly forms a rectangular box of volume $$2\pi r h \Delta x$$.
• Don't slice it the way you would normally slice this cake -- that doesn't form slices that are useful for calculus.

When the students actually eat part of a shell or part of a washer, they are more likely to remember the different shapes and what they mean.

Remember to bring something like a filet knife if you can; cutting the shells can be a little tricky. But you can do it on the document camera, so it's pretty compelling.

• Love it: I just usually bring in the bundt cake pan itself. – kcrisman Jan 14 '19 at 14:22
• And then you pull out the industrial saw to cut up the pan? ;) – Chris Cunningham Jan 14 '19 at 14:50
• LOL no they just have to imagine it, but that would be a very effective demonstration. – kcrisman Jan 14 '19 at 20:54
• I'd say you should take the time to bake them a cake; it's trivially easy but they think it is hard. You get probably 10-15 hours' worth of appreciation from the students for 1 hour of effort. – Chris Cunningham Jan 14 '19 at 22:02
• Haven't taught this material in five years, and not likely to in the immediate future but I agree with the spirit! I still remember a physics prof in college making a capacitor out of a garbage bag - though you can't eat that. – kcrisman Jan 15 '19 at 14:50