Example 8. Find the surface integral

$$
I=\int_{\Sigma}\left(\varphi \frac{\partial \psi}{\partial n}-\psi \frac{\partial \varphi}{\partial n}\right) d \sigma
$$

taken over the surface $\Sigma: x^{2}+y^{2}=R^{2}, z=0, z=H(H>0)$, if $\varphi=x^{2}+y^{2}+x+z, \psi=x^{2}+y^{2}+2 z+x$.
Solution. The required integral by the second Green's formula is

$$
I=\iiint_{V}(\varphi \Delta \psi-\psi \Delta \varphi) d v
$$

For the given functions $\varphi$ and $\psi$, we have $\Delta \varphi=4, \Delta \psi=4$, and thus,

$$
I=-4 \iiint_{V} z d u
$$

Transitioning to cylindrical coordinates $x=\rho \cos \varphi, y=\rho \sin \varphi, z=z$, we get:

$$
I=-4 \iiint_{V} z \rho d \rho d \varphi d z=-4 \int_{0}^{2 \pi} d \varphi \int_{0}^{R} \rho d \rho \int_{u}^{\|} z d z=-2 \pi R^{2} H^{2}
$$