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:

$${\int }_S{(\sin u\cos v-\sin v)}^2{d}^{2}A={\int }_0^{2\pi }{\int }_0^{\pi }{(\sin u\cos v-\sin v)}^2\sin ududv=$$

$${\int }_0^{2\pi }{\int }_0^{\pi }\left({\sin }^{3}u{\cos }^{2}v-2{\sin }^{2}u\sin v\cos v+\sin u{\sin }^{2}v\right)𝑑u𝑑v=$$

$${\int }_0^{2\pi }\left(2{\cos }^{2}v-\frac{2}{3}{\cos }^{2}v-\pi \sin v\cos v+2{\sin }^{2}v\right)𝑑v=$$

$$2\pi -\frac{2\pi }{3}-0+2\pi =\frac{10\pi }{3}$$

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