Paraboloid

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

$$z=a(x^2+y^2),$$ (1)

or parametrically by

$$x(u,v) = \sqrt{u}\,\cos v$$ (2)

$$y(u,v) = \sqrt{u}\,\sin v$$ (3)

$$z(u,v) = u,$$ (4)

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

The Volume of the paraboloid is

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

The weighted mean of $z$ over the paraboloid is

$$\left\langle{z}\right\rangle{}=\pi \int_0^h z^2\,dz={\textstyle{1\over 3}}\pi h^3.$$ (6)

The Centroid is then given by

$$\bar z={\left\langle{z}\right\rangle{}\over V}={\textstyle{2\over 3}}h$$ (7)

(Beyer 1987).

See also Elliptic Paraboloid, Hyperbolic Paraboloid, Parabola

References

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.