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 Incoming links: Bicylinder

At a fixed

z \in [- 1, 1]

∣ x ∣ \leq 1 - z^{2}, ∣ y ∣ \leq 1 - z^{2} .

Thus the cross section is a square of side

2 1 - z^{2}

and area

4 (1 - z^{2})

. Integrating these cross-sectional areas gives the volume of this bicylinder:

Vol (R_{1}) = \int_{- 1}^{1} 4 (1 - z^{2}) d z = 4 [z - \frac{z ^{3}}{3}]_{- 1}^{1} = \frac{16}{3} .

The common region inside equal-radius circular cylinders whose axes meet orthogonally. Two cylinders give a bicylinder; three give a tricylinder.