Area of a hyperbolic paraboloid contained within a cylinder

[Juggler123]

I've posted on this before and have now realized 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$$\theta$$dz

I also know that 0$$\leq$$$$\theta$$$$\leq$$2$$\pi$$

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.