Surface Integral on a Surface Defined over a General 2-D Region

$\int {\int }_{S}f \mathit{\ⅆ}\mathrm{\σ} \={\int }_{x\=a}^{x\=b}{\int }_{y\=u\left(x\right)}^{y\=v\left(x\right)}f\left(x\,y\,z\left(x\,y\right)\right) \sqrt{1\+z_x^2\+z_y^2} \mathit{\ⅆ}y \mathit{\ⅆ}x$

$\int {\int }_{S}f \mathit{\ⅆ}\mathrm{\σ}\=$${\int }_{y\=a}^{y\=b}{\int }_{x\=u\left(y\right)}^{x\=v\left(y\right)}f\left(x\,y\,z\left(x\,y\right)\right) \sqrt{1\+z_x^2\+z_y^2} \mathit{\ⅆ}x \mathit{\ⅆ}y$