The volume where two pipes cross

Question

Two infinitely long solid cylinders, each of radius 1, intersect at right angles so that their axes cross.

What is the volume of the region that lies inside both cylinders? Give an argument that does not need integration, by comparing the solid with a sphere of radius 1.

Accepted Answer

16/3, about 5.33. Slicing. Let the two axes be the x-axis and the y-axis, so both cylinders are "horizontal". Slice the solid with a horizontal plane at height z, where z runs from minus 1 to 1. In that plane, the first cylinder (axis along x) contributes the strip where y is between minus s and s, with s = the square root of (1 minus z squared); the second cylinder (axis along y) contributes the strip where x is between minus s and s. The common region is a square of side 2s, with area 4(1 minus z squared). Compare with the sphere. Slice the unit sphere with the same plane at height z. The s…

The volume where two pipes cross

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