Let two long circular cylinders, of diameter D, intersect in such a way that their symmetry axes meet perpendicularly. Let each of these axes be horizontal, and consider the "room" above the plane that contains these axes, common to both cylinders. (In architecture this room is called a "cross vault".) The floor of the cross vault is a square of side D, and the ceiling consists of four curvilinear triangles, meeting at the top, and intersecting in arcs that come down from the topmost point to the vertices of the square. Problem: Calculate the volume of the cross vault. Amazingly, the painter Piero della Francesca managed to do this in the 1400s, long before the invention of calculus!
Hi; Here's some help with the setup that will give you an integral to evaluate.
Let z represent the height above the floor of the cross vault. Let x and y be coordinates along the axes of each cylinder. That is, the floor of the cross vault is a square in the xy plane with corners at the points (0,0,0), (D,0,0), (0,D,0), and (D,D,0). The "peak" of the vault is a point at the center of the square ...
This solution is comprised of a detailed explanation to calculate the volume of the cross vault.