# 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

 $\int_{S}x^{2}\,d^{2}A=2\int_{x^{2}+y^{2}

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

 $2\int_{-r}^{+r}\int_{-\sqrt{r^{2}-y^{2}}}^{+\sqrt{r^{2}-y^{2}}}\frac{rx^{2}}{% \sqrt{r^{2}-x^{2}-y^{2}}}\,dx\,dy=$
 $2r\int_{-r}^{+r}\left(\left.-\frac{1}{2}x\sqrt{r^{2}-x^{2}-y^{2}}+\frac{1}{2}(% r^{2}-y^{2})\arcsin\frac{x}{\sqrt{r^{2}-y^{2}}}\right)\right|_{-\sqrt{r^{2}-y^% {2}}}^{+\sqrt{r^{2}-y^{2}}}\,dy=$
 $2r\int_{-r}^{+r}\frac{\pi}{2}(r^{2}-y^{2})\,dy=\frac{4}{3}\pi r^{4}$