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Surface Integral Help
The figure shows the surface created when the $\displaystyle cylinder\ y^2 + z^2 = 1\ intersects\ the\ cylinder\ x^2 + z^2 = 1\ Find\ the\ area\ of\ this\ surface.$
I think to find the surface integral we need to do the double integral of ru cross rv, with respect to du and dv respectively. However, im not even sure how to get to that point.

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How about doing the red part below and multiplying by 16 right? You could work through the cross product but that's not necessary. Just use the formula:
$\displaystyle S=\mathop\int\int\limits_{\hspace{15pt}R} \sqrt{(f_x)^2+(f_y)^2+1} dy dx$
and that red one is $\displaystyle z=f(x,y)=\sqrt{1x^2}$ and you can figure out R by equating $\displaystyle 1y^2=1x^2$ so $\displaystyle y=x$. So it's just that triangular region in the xy plane above the red surface.

Quick question, where did you get 16 from? That is the answer, but im not seeing how you came up with that.

Well, you have 8 red pieces on top as you go around right? And then eight more on the bottom.