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Cavalieri’s Principle


Plane version: Suppose that two bounded plane figures have the following property: There exists a line such that any line parallel to it intersects each one of the given figures by two segments of equal length. Prove that the figures have equal areas.

Space version: Suppose that two bounded space bodies have the following property: There exists a plane such that any plane parallel to it intersects the two bodies by two figures of equal areas. Prove that the bodies have equal volumes.

Hint: An approach using integrals, solves this easily using the coarea formula.

A second approach is
Plane case: Take the approximations by trapezoids with height less than \frac{1}{n} for both polygons and take the limit.
Something similar can be done in the space.

Using this principle, find the common volume of the intersection of two infinite cylinders. (Suppose they intersect orthogonally and their symmetry axes also intersect.

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