Volume where two cylinders cross

MediumCalculus~6m

Two infinite solid cylinders, each of radius $1$ , have axes that intersect at right angles. Find the volume of the region contained in both cylinders (the Steinmetz solid). Give the value to three decimal places.

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Slice by horizontal planes $z = \text{const}$ . Each cylinder becomes a slab $|x|\le\sqrt{1-z^2}$ or $|y|\le\sqrt{1-z^2}$ - what shape is the overlap?
The cross-section is a square of side $2\sqrt{1-z^2}$ , area $4(1-z^2)$ . Integrate from $-1$ to $1$ .

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5.333 (± 0.01)

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Volume where two cylinders cross

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