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 \(\dfrac{{V}_{i}}{{V}_{r}}\).

Give your answer to 3 decimal places.


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