# example of integration over sphere with respect to surface area

As an example of how to use the formula derived in http://planetmath.org/node/6664example 1, let us consider the following example:

 $\int_{S}(\sin u\cos v-\sin v)^{2}d^{2}A=\int_{0}^{2\pi}\int_{0}^{\pi}(\sin u% \cos v-\sin v)^{2}\sin u\>du\,dv=$
 $\int_{0}^{2\pi}\int_{0}^{\pi}\left(\sin^{3}u\cos^{2}v-2\sin^{2}u\sin v\cos v+% \sin u\sin^{2}v\right)\>du\,dv=$
 $\int_{0}^{2\pi}\left(2\cos^{2}v-\frac{2}{3}\cos^{2}v-\pi\sin v\cos v+2\sin^{2}% v\right)\,dv=$
 $2\pi-\frac{2\pi}{3}-0+2\pi=\frac{10\pi}{3}$