Exploring Calculus with 3d printing

Last week we were exploring volumes of various solids in the calculus course I’m teaching my son. That subject is a great opportunity to use 3d printing to enhance the course.

In fact, this old video from Brooklyn Tech about 3d printing was one of my first exposures to using 3d printing in math education:

Over the last week we printed 3 different shapes. Tonight I used these shapes as props to go back and review some of the volume concepts we’ve been studying. The ideas were new to my son and he hasn’t quite mastered all of them yet, so the review was productive. The videos are a bit longer than usual as we review some of the concepts, though.

Here’s the first shape: y = Sin(x) revolved around the x-axis:

The next set of shapes were not volumes of curves rotated around the x-axis, but rather shapes where the slices were squares:

Finally, we looked at the curve y = \sqrt{x - 1} from x = 1 to x = 5 rotated around the y-axis. Here my son remembered how to calculate this volume by looking at the slices parallel to the x-axis, but struggled a bit with when the slices were cylindrical shells – so we spent a long time on that 2nd part.

I’m happy that we have the opportunity to explore these shapes with our 3d printer – it definitely is incredible to be able to hold shapes like these in your hand!