Help appreciated!
You want:
$\displaystyle
\int_S x^2+y^2 dS
$
where $\displaystyle S$ is the surface of a cylinder of radius $\displaystyle a$ and height $\displaystyle h$ (assume the origin is on the axis of the cylinder and the axis system is aligned in the natural manner with the axis of the cylinder, and that the bottom of the cylinder is at $\displaystyle z=0$). The area element in cylindrical polars is $\displaystyle a \,d\theta \, dz$. So, as $\displaystyle x^2+y^2=a^2$ on the cylinder, the integral is:
$\displaystyle
\int_0^{2 \pi} \int_0^h a^2 dz\, a\,d\theta = 2\, \pi \, h \, a^3
$
RonL