It’s one of my favorite weeks in Calculus: Volumes of Rotation Week!

I thought I would share my current collection of Volume demos. Today I’ll start with the site Demos with Positive Impact, build by David R. Hill and Lila F. Roberts. These are all demos that I have been using for YEARS now. They are absolutely wonderful, and the galleries include the functions in the examples – which makes them great for classroom examples. Start at the Visualizations for Volumes of Solids in Calculus page.
I always begin with Volumes by Section by using the examples in the Demo Gallery. Who can draw these? Here’s y=sin(x) with equilateral triangle cross sections:
Then I move on to the Disk and Washer Methods for Solids of Revolution. There are some great real-life examples of volumes of revolution and the disk method and washer method on the site.

Check out the Disk Method Demo Gallery and the Washer Method Demo Gallery for more than 10 examples that you could use. Some are revolved around the x-axis, and some around the y-axis. I particularly like the 4-part construction of the washer-method problems (as shown in this picture below).

The shell method has got to be the toughest to visualize, and here’s where these demos are invaluable! Several real-life examples… nested Russian dolls, Angel Food cakes, Combos, toilet paper rolls, etc. are shown on the Shell Method Page and then (as always) there are a great collection of mathematical solids (4-part construction) in the Gallery.

If you just want to jump quickly to all the Galleries of Demos, go here.

Now, I don’t have a computer lab for my calculus class, but I have used these materials in several different ways from year to year: 1) Schedule time in a computer lab so that students can see each demo up close, 2) use a projection system, 3) have the students that have laptops bring the site up on their laptops. Almost all my calculus students have laptops now, so we essentially have a functioning “lab” if they all share their laptops.