A Generalized Cavalieri-Zu Principle
Sidney Kung

# Volume of an Elliptic Paraboloid

Consider an elliptic paraboloid as shown below, part (a):

At $z=h$ the cross-section is an ellipse whose semi-mnajor and semi-minor axes are, respectively, $u$ and $v$. Since $u=b\sqrt{h}$, and $v=a\sqrt{h}$, $A=\pi abh$. We choose a triangular prism $PR$ of height $\pi ab$, and whose cross-section is an isosceles right triangle. At $z=h$, $A'=\pi abh=A$. Hence, from (**), the volume of $EP$ is

$\displaystyle V_{EP}=\frac{\pi abc^{2}}{2}$

If the paraboloid is defined by a more dimesionally balanced equation like, say,

$\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{z}{c}$

then the volume of the paraboloid at height h will be given by

$\displaystyle V_{EP}=\frac{\pi abh^{2}}{2c}$

and, for $h=c$,

$\displaystyle V_{EP}=\frac{\pi abc}{2}.$