# surface area using cylindrical coordinates

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 ?

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