Evaluate $\displaystyle \int\!\!\int_{S} \sqrt{ 1 + x^{2} + y^{2} } \: dS$ where $S$ is the helicoid: $\mathbf{r}(u, v) = u\cos(v)\mathbf{i} + u\sin(v)\mathbf{j} + v\mathbf{k}$, with $0 \leq u \leq 2, 0 \leq v \leq 3\pi$