One of our last topics in Calculus is volume of solids of revolution. I teach disc method first, then shell method. Between the two, we were desperately in need of something to mix up too many days of direct instruction and practice problems, so I wrote up this project during parent-teacher conferences.
Essentially:
- Find an object with a circular cross-section
- Measure it, photograph it, insert into Desmos and write functions to model its outline
- Write and calculate the necessary integrals for the solid of revolution
- Summarize your work in some format and compare your volume to the true volume of the object if possible
Lessons learned:
- Photograph objects carefully so that nothing is skewed
- MEASURE CAREFULLY! and then MEASURE AGAIN!!!
- Be sure to CENTER the object on the axis in Desmos!
- I’m okay with students using Wolfram Alpha for the integral calculations because by hand takes FOREVER with those equations from regression models.
- Remind students what a piecewise function should look like.
- Have a measuring cup/beaker handy to measure the true volume.
Overall, I thought this was a pretty cool project, although a few of my students had some error due to issues listed above. One student got within ONE ml of the true volume though, which was pretty amazing!
1ntegration by Parts 