For $a>0$ I'm trying to calculate the surface of area with given equations $x^2 + y^2 \leq a$ and $az = xy$.
I think it should be done with $$\iint_D |\vec{r_u} \times \vec{r_v}| \,du\,dv $$, which is the surface of $\Sigma$ with the regular parametrization $\vec{r}: D \rightarrow \Sigma, \vec{r}(u,v) = (X(u,v), Y(u,v), Z(u,v))$.
I get that the surface is (if I'm not wrong) an intersection of a cylinder and a hyperbolic paraboloid, but kind of got stuck when trying to find a parametrization of the surface. Any hints would be very appreciated, thank you! :)