anonymous 4 years ago Evaluate the surface integral $$\int\int_{\sigma}\sqrt{2e^{2(x+y)}+1}\text{ dS}$$ given that S is the surface $$z=e^{x}e^y$$ above the triangular region in the xy-plane enclosed by $$y=1-x$$ and the coordinate axes.

Here's our region D|dw:1329748292145:dw|so our bounds will be$0\le y\le1-x$$0\le x\le 1$now we need dS, and since this surface is given by z=g(x,y) our formula for that will be$dS=\sqrt{(\frac{dz}{dx})^2+(\frac{dz}{dy})^2+1}dA=\sqrt{(e^xe^y)^2+(e^xe^y)^2+1}dA$$dS=\sqrt{2e^{2(x+y)}+1}dA$which is extremely convenient, because now our integral will be much easier$\int\int_\sigma\sqrt{2e^{2(x+y)}+1}dS=\int_{0}^{1}\int_{0}^{1-x}2e^{2(x+y)}+1dydx$you should be able to integrate this yourself without too much trouble.