A Generalized Cavalieri-Zu Principle
Sidney Kung

# Volume of a Hyperboloid of One Sheet

A hyperboloid of one sheet is the surface obtained by revolving a hyperbola around its minor axis. Denote the solid bounded by the surface and two planes $y=\pm h$ by $H$. At the level $d$ above the $x$-axis, the cross-section of $H$ is a circle of radius $\displaystyle \frac{a}{b}\sqrt{b^{2}+d^{2}}$. So, $\displaystyle A_{1}=\frac{\pi a^2}{b^2}(b^{2}+d^{2})$.

We place two solids $P$ and $W$ parallel to $H$ on the $x$-axis. Their dimensions are chosen such that $A_{2}=\pi (b+d)^{2}$ and $A_{3}=2\pi bd$. Thus, $\displaystyle A_{1}=\frac{\pi a^{2}}{b^2}[(b+d)^{2}-2bd]=\frac{a^2}{b^2}(A_{2}-A_{3})$. Hence, by (**), the volume of the hyperboloid $H$ is $\displaystyle 2\times\frac{a^2}{b^2}[\mbox{volume of}\space P-\mbox{volume of}\space W\space ]$, which is

$\displaystyle V_{H}=\frac{2a^2}{b^2}\{\frac{\pi h}{3}[(b+h)^2+b(b+h)+b^{2}]-\pi h^{2}b\}=\frac{2\pi ha^2}{b^2}\left(b^{2}+\frac{h^2}{3}\right).$  Copyright © 1996-2018 Alexander Bogomolny

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