This is part three in a series about teaching volumes in calculus. The first post can be found here.
My students return for day three of volumes. They have spend the last night trying to figure out how high up the glass the Pepsi will go. Some of my students have random guesses, some have looked up volume equations for frustums, others have tried to use volumes by cross sections and proportions to make their guesses.
Then I put up my guess, with three decimal places. They know I've done some crazy math, but they're not quite sure what I did. I hold their interest while I pour, and believe it or not, I'm the closest.
I show them my work. I have set up an equation for one the side of the glass, I explain how I revolved the line, created disks, and then put the integral and the volume I want into my calculator and found the intersection of the two.
I expand this discussion of rotations and we pull our our Play-Doh again. Before rotating the line y equals x squared around the x-axis, the students practice visualizing it:
I've found the Play-Doh to be the best thing that's happened to my teaching of volumes. Not only does it force all of the students to solidify their mental picture (pun intended), it also gives me a great way to quickly evaluate where my students are in their understanding.
Next up: The final project that ties everything together.