example of integration with respect to surface area of a paraboloid
In this example we examine the paraboloid given by the equation $z={x}^{2}+3{y}^{2}$. Putting $g(x,y)={x}^{2}+3{y}^{2}$, we have
$$\sqrt{1+{\left(\frac{\partial g}{\partial x}\right)}^{2}+{\left(\frac{\partial g}{\partial y}\right)}^{2}}=\sqrt{1+{\left(2x\right)}^{2}+{\left(6y\right)}^{2}}=\sqrt{1+4{x}^{2}+36{y}^{2}}$$ 
and hence
$${\int}_{S}f(x,y){d}^{2}A=\int f(x,y)\sqrt{1+4{x}^{2}+36{y}^{2}}\mathit{d}x\mathit{d}y.$$ 

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Title  example of integration with respect to surface area^{} of a paraboloid 

Canonical name  ExampleOfIntegrationWithRespectToSurfaceAreaOfAParaboloid 
Date of creation  20130322 14:58:20 
Last modified on  20130322 14:58:20 
Owner  yark (2760) 
Last modified by  yark (2760) 
Numerical id  15 
Author  yark (2760) 
Entry type  Example 
Classification  msc 28A75 