Last spring, I wrote about a project I gave my students in second semester calculus to visualize solids of known cross section. (If you haven’t read that post yet, do it now; this post won’t make sense otherwise.) It was unexpectedly successful with my students, and so I have been pushing other teachers to try it. A couple of weeks ago, Mr. Ordonez, a new teacher in my department, agreed. His class was hesitant at first, but once they got started, they really appreciated having done it. The exciting part for me, though, is an observation Mr. Ordonez made afterward.
Mr. Ordonez gave a handful of different prompts to his students, so as an individual project, some of the students got the same prompt. By chance, two students who had gotten the same prompt, and a prompt in which the cross sections were semicircles no less, used the exact same scale (1 unit=1/2 inch) in their construction. Mr. Ordonez noticed that he could place them back-to-back to get a solid of revolution.
His observation opens up entirely new possibilities for future projects. Perhaps a version of this project could involve volumes of revolution where the cross sections are genuine washers with a hole in the middle. Is it possible to adapt the project to demonstrate the shell method? Let me know in the comments section. I’d love to see your pictures!