In this example we examine the paraboloid given by the equation $z = x^2 + 3 y^2$ . Putting $g(x,y) = x^2 + 3 y^2$ , we have $$\sqrt{1 + \left( \frac{\partial g}{\partial x} \right)^{\!2} + \left( \frac{\partial g}{\partial y} \right)^{\!2}} = \sqrt{1 + \left( 2 x \right)^2 + \left( 6 y \right)^2} = \sqrt{1 + 4 x^2 + 36 y^2 }$$ and hence $$\int_S f(x,y) \, d^2 A = \int f(x,y) \sqrt{ 1 + 4 x^2 + 36 y^2 } \, dx \, dy.$$

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