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The Surface of Revolution of the Parabola. It is a Quadratic Surface which can be specified by the Cartesian equation

\end{displaymath} (1)

or parametrically by
$\displaystyle x(u,v)$ $\textstyle =$ $\displaystyle \sqrt{u}\,\cos v$ (2)
$\displaystyle y(u,v)$ $\textstyle =$ $\displaystyle \sqrt{u}\,\sin v$ (3)
$\displaystyle z(u,v)$ $\textstyle =$ $\displaystyle u,$ (4)

where $u\in[0,h]$, $v\in [0,2\pi)$, and $h$ is the height.

The Volume of the paraboloid is

V=\pi \int_0^h z\,dz={\textstyle{1\over 2}}\pi h^2.
\end{displaymath} (5)

The weighted mean of $z$ over the paraboloid is
\left\langle{z}\right\rangle{}=\pi \int_0^h z^2\,dz={\textstyle{1\over 3}}\pi h^3.
\end{displaymath} (6)

The Centroid is then given by
\bar z={\left\langle{z}\right\rangle{}\over V}={\textstyle{2\over 3}}h
\end{displaymath} (7)

(Beyer 1987).

See also Elliptic Paraboloid, Hyperbolic Paraboloid, Parabola


Beyer, W. H. (Ed.) CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 133, 1987.

Gray, A. ``The Paraboloid.'' §11.5 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 221-222, 1993.

© 1996-9 Eric W. Weisstein