How do I compute the surface area of the following surface?

I need to compute the surface area of the following surface:

$z=xy$ and $x^2+y^2\leq 1$

I'm not really sure how to procede, so I know that $A(\phi)=\int_Q \sqrt{g_\phi(x)}dx$ where $g_\phi(x)$ is the determinant of the gram matrix. But to get to this point I first need to find $\phi$. I thought to use the parametrisation $(x,y)=(r\cos(\theta),r\sin(\theta))$. And then define $$\phi:[0,1]\times [0,2\pi]\rightarrow \mathbb{R}^3\,\,\,\,\,\,\,\,\,\, \phi(r,\theta)=(r\cos(\theta),r\sin(\theta),r^2\cos(\theta)\sin(\theta))$$

Does this works?

Thank you for your help

Solution 1:

$v=(\cos\theta,\sin\theta, r\sin2\theta)$

$w=(-r\sin\theta,r\cos\theta,r^2 \cos2\theta)$

$ \displaystyle G=\left(\begin{array}{rrr} 1 + r^2 \sin^2 2\theta & \dfrac{r^3 \sin 4\theta}{2} \\ \dfrac{r^3 \sin 4\theta}{2} & r^2 + r^4 \cos^2 2\theta \end{array}\right)$

$ \text {Det } G = r^2 + r^4$

Can you take it from here?