$\displaystyle \displaystyle \begin{align*} \sqrt{(\frac{\partial z}{\partial x})^2 + (\frac{\partial z}{\partial x})^2 + 1}\ &=\ \sqrt{\frac{x^2}{x^2 + y^2}\ +\ \frac{x^2}{x^2 + y^2} + 1} \\ \\ &=\ \sqrt{2} \end{align*}$

and,

$\displaystyle \displaystyle \begin{align*} S\ &=\ \int \int _D \ \sqrt{(\frac{\partial z}{\partial x})^2 + (\frac{\partial z}{\partial y})^2 + 1}\ dA \\ \\ &=\ \int \int _D \ \sqrt{2}\ dA \\ \\ &=\ \int \int _P \ \sqrt{2}\ r\ dr\ d\theta \\ \\ &=\ \int _0^{\pi} \int _0^{2 \cos \theta} \ \sqrt{2}\ r\ dr\ d\theta \end{align*}$

... where (key in spoiler) ...