Calculus Graphics

Here are a few snapshots of a demonstration I like. The problem involved is the standard one of determining the volume of the intersection of two cylinders. The difficulty with this problem is, in part, figuring out what this object looks like. I have demonstration in AVS which solves this problem. It starts with a picture of the two cylinders. In AVS I can make the two cylinders transparent. Increasing the transparency slowly reveals the intersection of the two cylinders. This object can then be rotated in 3-dimensions in AVS, so students can get a very good idea what it looks like. The demonstration goes on to show how to cut the object up into a stack of squares (shown here with the intersection of the cylinders being only partly transparent). We review the general principle for computing volumes and, after some work at the blackboard and interactive discussion of how to compute the cross-sectional area we need (assuming the cylinders have radius 1 unit), we are able to compute the volume of the object.

The various 3-dimensional objects were constructed by describing them as objects in Mathematica, then importing them to AVS. All the pictures available here are only half the size of the ones used in the demonstration. Some of them suffer for this size reduction.

Exercise: The intersection of the two cylinders has four curved ``faces''. Each can be flattened out, so the shape could be cut out of paper four times to make a model, for instance. Describe the flattened out shape of one face.