# 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

 $\int_{S}f(x,y)\,d^{2}A=\int f(x,y)\sqrt{1+4x^{2}+36y^{2}}\,dx\,dy.$
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Title example of integration with respect to surface area of a paraboloid ExampleOfIntegrationWithRespectToSurfaceAreaOfAParaboloid 2013-03-22 14:58:20 2013-03-22 14:58:20 yark (2760) yark (2760) 15 yark (2760) Example msc 28A75