The question is
Let $r>0$. Let S⊂ $R^3$ the surface given by $y_1y_2-ry_3=0$. Determine the area of the portion of S inside the cylinder given by $y_1^2+y_2^2=r^2$.
In class we saw a formula to calculate the area of parametric surfaces (the one with the cross product). The problem is, I don't have a parametrisation of S. And to be very honest, I have no idea how it's done. I thought maybe cylindrical coordinates would help? I also tried isolating r to see if it could help in any way. How do I approach a question like this? We didn't see any examples in class and I am very confused.
You need to find surface area of the hyperbolic paraboloid $z = \frac{xy}{r}$, inside the cylinder $x^2+y^2 = r^2$.
The formula for the surface area is given by,
$S = \displaystyle \iint_A \sqrt{1+ \bigg(\frac{\partial z}{\partial x}\bigg)^2 + \bigg(\frac{\partial z}{\partial y}\bigg)^2} \ dA$
$ = \displaystyle \iint_A \sqrt{1+ \frac{x^2}{r^2} + \frac{y^2}{r^2}} \ dA$
where $A$ is the projection of the surface in $XY$ plane and $dA = dx \ dy$.
Converting to polar coordinates,
$x = \rho \cos \theta, y = \rho \sin \theta, 0 \leq \rho \leq r, 0 \leq \theta \leq 2\pi$.
$S = \displaystyle \frac{1}{r}\int_0^{2\pi} \int_0^r \rho \sqrt{\rho^2 + r^2} \ d\rho \ d\theta = \displaystyle \frac{2 \pi}{r} \int_0^r \rho \sqrt{\rho^2 + r^2} \ d\rho $
To evaluate the integral, you can substitute $\rho^2 + r^2 = t, \rho d\rho = \frac{1}{2} dt$.
Can you take it from here?