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.
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.