Calculating the volume of two intersecting cylinders
Consider two cylinders which intersect each other. What is the volume
of their intersection?
Use Maple to provide a method to calculate the
volume of the intersection of two arbitrary
cylinders. This is not as easy as it might seem, so it is a good idea
to start with the simpler problem of finding the area of
intersecting cylinders assuming that the cylinders are at right angles
to each other, have the same radius and that their axes lie in the
Naturally, you should use multiple integration to calculate
this volume and the task is much simplified by noticing that choosing
crosssections in a certain orientation results in rectangular regions.
(How does this change if the restriction that the cylinders are
perpendicular to each other is relaxed?) The areas of rectangles are
easy to calculate, so the the whole problem in this simple case is in
determining the exact dimensions of the crosssectional rectangles.
Next, generalize your method so that you do not have to assume that the
cylinders are perpendicular to each other; then,
relax the assumption that the cylinders have the same radius.
Finally, solve the problem without even assuming that the axes
of the cylinders lie in the same plane.
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