Calculus art project

Day 4 of #MTBoS30--May the 4th be with you!

My calculus class and I have been looking at the written response problems from AP Calculus.  This is not an AP class, but I like to use the problems as they do a good job of integrating all skills the students should know and understand.  I model my final exam after the AP test.

So we are working on the area and volume problems, specifically those ones that say "the base of a figure consists of the area between the curves f(x) = sin x and g(x) = -x² + 3x + 2.  The figure is composed of equilateral triangles perpendicular to the x axis. Find the volume of the figure."
Every time we go over these problems, I try to draw some sort of perspective drawing on the board and they all just come out looking like weird blobs.  A couple of times, I have cut out a triangle (or half circle, or flying monkey or whatever figure they are using) to demonstrate to my students.  I'm not sure anybody has ever gotten the idea of one of these.
Yesterday, we got one involving squares, so I decided to have my students draw the function on paper and then cut out squares and try to create the figure.  I saw this in a tweet or a blog recently and thought it was a good idea.
      The hardest thing was to get the squares to stand up so that you could picture what the finished figure would look like.  We did 10 squares to try to show the object.  Once we had the object created using squares, we stood them on the desks and looked at them from different angles to try to picture what the solid would look like.  One thought I had was to try to cover it with paper to show the surface of the solid.  Not sure my paper cutting skills are up to that.


I wish I had taken photos of our drawings.  They were somewhat "abstract" still.  My thought for this in the future is to cut out squares of wood then clamp them together and sand the surface smooth so that you could show the students the component parts, then assemble the squares together to get a better idea of what the solid will really look like.  I'm not sure my woodworking skills are up to this either, but I might try this summer.

Has anybody else tried this?  Or does anybody know of a 3-d graphing program that could model this type of problem?  I could see this helping out on the volume problems where you rotate a function around a line also.  Those problems seem to be easier to visualize than the "object perpendicular to the x axis on a base defined by two functions" type.
Also wondering if this is a completely artificial problem constructed just for the likes of AP exams or are there places where you might model a real world object using this technique?  (In my mind, I'm picturing boat design, but that is the only thought I have and I don't know if it applies.)