First, I bought a package of forty 5.5x8.5" foam sheets that were self-adhesive on one side ($5 at Wal-Mart). I stacked three sheets together (so I have thirteen "boards") to produce the base of the desired solid.
Then, I graphed two functions (y=2cos(x/2) and y=e^(x/4)) on Desmos, printed them off, and used them as stencils. So, each foam board has a graph on both sides:
I don't really know why I chose these two graphs other than the fact that I wanted one increasing and one decreasing function, both only in the first quadrant.
Next, I made and printed different kinds of cross sections for the kids to use on cardstock (see file below: squares, rectangles, semi-circles, equilateral triangles, and isosceles right triangles.
And after that, the students did the rest of the work. They worked in groups of 2-3 to create a solid with either base f(x) or g(x) (I assigned). Then, they calculated the volume of their solid and put their answer in this table:
Here are some examples of their finished products:
Here's how I told the kids to use the pins:
Finally, here is everything you'll need if you want to do this with your calc kids, too!
The first two pages are the two graphs I used. The next five pages are the cross sections that I printed off on cardstock. The graphs/cross sections are sized to fit together. All you'll need is some foam sheets, pins, and Sharpies. :)