The surface [tex]S[/tex] can be parameterized by
[tex]\mathbf r(\theta,z)=(2\cos\theta,2\sin\theta,z)[/tex]
where [tex]0\le\theta\le2\pi[/tex] and [tex]0\le z\le2[/tex]. Then the surface integral can be computed with
[tex]\displaystyle\iint_S(x^2+y^2+z^2)\,\mathrm dS[/tex]
[tex]\displaystyle=\int_{\theta=0}^{\theta=2\pi}\int_{z=0}^{z=2}(4\cos^2\theta+4\sin^2\theta+z^2)\|\mathbf r_\theta\times\mathbf r_z\|\,\mathrm dz\,\mathrm d\theta=\dfrac{128\pi}3[/tex]
