multivariable calculus - Surface integral of a function

Evaluate the integral
$\iint_S\frac{x^{2}+y^{4}z}{x^{2}+y^{2}+z^{2}}dS$
where
$$S=\{(x,y,z)\;:\; x^{2}+y^{2}=1 ,-1\leq z\leq 1\}.$$

Can someone please show me how to calculate this?

And what is it am calculating, I am new on this topic.

I can do questions of this type :
Surface integral
But I don't even know what is it am calculating here.

I know I should parametrize my region $S$, it is a circular cylinder of radius $1$ and goes from $z=-1$ to $z =1$.

Thanks in advance.

Answer

Hint. Use the cylindrical coordinates with $dS=r d\theta dz$.

Here $r=1$, $x=\cos\theta$, $y=\sin \theta$.
Hence the integral can be written as
$$\int_{\theta=0}^{2\pi} \int_{z=-1}^1 \frac{\cos^{2}(\theta)+\sin^{4}(\theta)z}{1+z^{2}} dz d\theta.$$
Note that since $S$ is symmetric with respect to the plane $z=0$, the term $\sin^{4}(\theta)z$ can be removed.