Area of a hyperbolic paraboloid contained within a cylinder

[Juggler123]

I've posted on this before and have now realised I was doing it completely wrong before but's still bugging me. I have to find the area of the hyperbolic paraboloid z=xy contained within the cylinder x^{2}+y^{2}=1.

I've parametrized x^{2}+y^{2}=1 into polar coordinates to give that dA=d\thetadz

I also know that 0\leq\theta\leq2\pi

But I'm still having trouble finding the z limits for that part of the integration. Any help would be great. Thanks.

[LCKurtz]

It isn't the cylinder you need to parameterize, it's the hyperbolic paraboloid. Incidentally, that part inside the cylinder looks just like a pringle. Try this parameterization:

\vec{R}(r,\theta) = \langle r*cos(\theta), r*sin(\theta),r^2cos(\theta)sin(\theta)\rangle

with
dS = |\vec{R}_r \times \vec{R}_\theta |dr d\theta

where (r, \theta) are the usual polar variables in the xy plane.