Surface Integrals, Stokes & Divergence · Challenge problem

The helicoid $S$ is given by $\mathbf r(u, v) = (u\cos v,\ u\sin v,\ v)$ for $0 \le u \le 1$, $0 \le v \le \pi$. Compute $\iint_S \sqrt{1 + x^2 + y^2}\,dS$. Give an exact answer in terms of $\pi$.