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 two intersecting cylinders assuming that the cylinders are at right angles to each other, have the same radius and that their axes lie in the same plane. 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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