example of integration with respect to surface area of a paraboloid

In this example we examine the paraboloid given by the equation $z=x^2+3y^2$. Putting $g(x,y)=x^2+3y^2$, we have

$$\sqrt{1+{\left(\frac{\partial g}{\partial x}\right)}^2+{\left(\frac{\partial g}{\partial y}\right)}^2}=\sqrt{1+{\left(2x\right)}^2+{\left(6y\right)}^2}=\sqrt{1+4x^2+36y^2}$$

and hence

$${\int }_{S}f(x,y)d^{2}A=\int f(x,y)\sqrt{1+4x^2+36y^2}𝑑x𝑑y.$$

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