Assignment:
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!
Provide complete and step by step solution for the question and show calculations and use formulas.