$dS = \sqrt{1 + 4x^2 e^{-2(x^2+y^2)} + 4y^2 e^{-2(x^2+y^2)}} \, dx \, dy$

$dS = e^{-(x^2+y^2)} \sqrt{1 + 4(x^2+y^2)} \, dx \, dy$

$dS = \sqrt{1 + (\frac{\partial f}{\partial x})^2 + (\frac{\partial f}{\partial y})^2} \, dx \, dy$

$dS = \sqrt{1 + e^{-2(x^2+y^2)}} \, dx \, dy$