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I'm going a little crazy to be able to figure this out. I have the surface given by the $z = xy$ plane inside the cylinder $x^2+ y^2=1$. When I do the integral by Cartesian coordinates and then converting to polar.

$$ A = \int\int\sqrt{1 + (\partial z/\partial x)^2 + (\partial z/\partial y )^2}\,dx \, dy . $$ EDIT: $$ A = \int_0^{2\pi}\int_0^1\sqrt{1 + \rho ^2}\,\rho d\rho d\theta . $$ $$ A = \frac{2\pi}{3}(2 \sqrt{2} - 1). $$ When I try to solve it by cylindrical coordinates I do not get the same result. How could i get it using cylindrical coordinates ?

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Hint: $\quad dxdy=\rho d\rho d\theta$.

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