Example7OfIntegrationWithRespectToSurfaceArea 2005-01-27 16:35:41 2006-06-14 14:42:12 Example surface integration with respect to area 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 $$\int_S f(x,y) \, d^2 A = \int f(x,y) \sqrt{ 1 + 4 x^2 + 36 y^2 } \, dx \, dy.$$ \PMlinkescapetext{\sl Quick links:} \begin{itemize} \item \PMlinkid{main entry}{6660} \item \PMlinkid{previous example}{6669} \item \PMlinkid{next example}{6673} \end{itemize}