As an example of how to use the formula derived in example 1, let us consider the following example: $$\int_S (\sin u \cos v - \sin v)^2 d^2 A = \int_0^{2 \pi} \int_0^\pi (\sin u \cos v - \sin v)^2 \sin u \> du \, dv =$$ $$\int_0^{2 \pi} \int_0^\pi \left( \sin^3 u \cos^2 v - 2 \sin^2 u \sin v \cos v + \sin u \sin^2 v \right) \> du \, dv =$$ $$\int_0^{2 \pi} \left( 2 \cos^2 v - \frac{2}{3} \cos^2 v - \pi \sin v \cos v + 2 \sin^2 v \right) \, dv =$$ $$2 \pi - \frac{2 \pi}{3} - 0 + 2 \pi = \frac{10 \pi}{3}$$

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