# Generalized Chinese Puzzle

Calculus Level 5

Consider a body that is formed by the intersection of two identical and perpendicular cylinders. The cross-section of the cylinders has at least one axis of symmetry. Also, the boundary of the half of the cross section can be expressed as a function of this axis (the half cross section is the area under the function).

Now consider a second volume created by the revolution of the half cross section area. What is the ratio of the intersection volume to the revolution volume?

For example:

Take any function (it can be piecewise).

Create the cylinders.
We get the intersection volume $${V}_{i}$$
We also need the revolution volume $${V}_{r}$$
And calculate the ratio $$\dfrac{{V}_{i}}{{V}_{r}}$$.