The volume of the intersection of three cylinders

Begin with the simpler case of three orthogonal cylinders, all of the same radius. The key obervation you will have to make is that, whereas, in the case of two cylinders the intersection along a suitably chosen plane is always a square. in the case of three cylinders it will be either a square or a square with rounded corners. The square with rounded corners will occur when the square formed by the two cylinders whose axes are parallel to the plane no longer fits in the circle formed by the third cylinder. The accompanying diagram illustrates this. The red circle represents the crosssection of a cylinder perpendicular to the page, while the square is formed by the intersection of the two cylinders parallel to the page. Note that if the square is so small that its corners are contained in the circle then the intersection of the square and circle is ismple a square. Hence, in order to solve this problem you will have to deal with the two cases. You may decide to break up the integral into two parts, according to the cases.

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