computation of moment of spherical shell
In using the formula for area integration over a sphere derived in the http://planetmath.org/node/6668last example, we need to keep in mind that to every point in $xy$ plane, there correspond two points on the sphere, which are obtained by taking the two signs of the square root. The importance of this fact in obtaining a correct answer is illustrated by our next example, the calculation of the moment of inertia of a spherical shell.
The moment of a spherical shell is given by the integral
$$I={\int}_{S}{x}^{2}{d}^{2}A.$$ 
While we could compute this by first converting to spherical coordinates^{} and then using the result of http://planetmath.org/node/6664example 1, we can avoid the trouble of changing coordinates by treating the sphere as a graph. Using the result of the previous example, our integral becomes
$$ 
where the factor of 2 takes into account the observation of the preceding paragraph that two points of the sphere correspond to each point of the $xy$ plane. Computing this integral, we find
$$2{\int}_{r}^{+r}{\int}_{\sqrt{{r}^{2}{y}^{2}}}^{+\sqrt{{r}^{2}{y}^{2}}}\frac{r{x}^{2}}{\sqrt{{r}^{2}{x}^{2}{y}^{2}}}\mathit{d}x\mathit{d}y=$$ 
$${2r{\int}_{r}^{+r}\left(\frac{1}{2}x\sqrt{{r}^{2}{x}^{2}{y}^{2}}+\frac{1}{2}({r}^{2}{y}^{2})\mathrm{arcsin}\frac{x}{\sqrt{{r}^{2}{y}^{2}}}\right)}_{\sqrt{{r}^{2}{y}^{2}}}^{+\sqrt{{r}^{2}{y}^{2}}}dy=$$ 
$$2r{\int}_{r}^{+r}\frac{\pi}{2}({r}^{2}{y}^{2})\mathit{d}y=\frac{4}{3}\pi {r}^{4}$$ 
Quick links:

•
http://planetmath.org/node/6660main entry

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http://planetmath.org/node/6668previous example
Title  computation of moment of spherical shell 

Canonical name  ComputationOfMomentOfSphericalShell 
Date of creation  20130322 14:58:11 
Last modified on  20130322 14:58:11 
Owner  rspuzio (6075) 
Last modified by  rspuzio (6075) 
Numerical id  13 
Author  rspuzio (6075) 
Entry type  Example 
Classification  msc 28A75 