Evaluate $\displaystyle \int\int_{{S}}\sqrt{{{1}+{x}^{{2}}+{y}^{{2}}}}{d}{S}$, where $\displaystyle {S}$ is the helicoid with vector equation $\displaystyle {r}{\left({u},{v}\right)}=<{u}{\cos{{v}}},{u}{\sin{{v}}},{v}>\quad{0}\le{u}\le{1},\ {0}\le{v}\le\pi$