Finding the Area of a Hyperbolic Paraboloid

[Juggler123]

I need to find the area of the hyperbolic paraboloid z=xy contained within the cylinder x^2+y^2=1. I know I need to take a double integral but am having real difficulty finding the correct limits, so far I've got that;

$$\int dx$$$$\int dy$$

With the x limits being 1 and -1 and the upper y limit to be sqrt(1-x^2) I'm having trouble finding the lower y limit. Although to be honest I'm not completely sure about the other three limits! Sorry about my awful attempt at Latex-ing I don't know how to do it so couldn't write the limits of the integrals on the integral. Any help would be great! Thanks.

 

[mikeph]

the problem is symmetric by pi/2 so I'd just stick to the (+,+) quadrant and your lower limit is y=0. Then your x limits would be 1 and 0.

The area in the entire domain is then four times the area in a single quadrant.

 

[HallsofIvy]

I have no idea what you mean by

so far I've got that;

$$\int dx$$$$\int dy$$

Shouldn't there be some function to be integrated in that? And it probably is NOT
$$\int dx\int dy$$
but rather
$$\int f(x,y) dxdy$$
Even ignoring the "f(x,y)" the two separate integrals implies that the two coordinates can be separated- which is not the case here- at least not in Cartesian coordinates.

The surface area of z= f(x,y) is given by
$$\int\int \sqrt{1+ \left(\frac{\partial f}{\partial x}\right)^2+ \left(\frac{\partial f}{\partial y}\right)^2} dA$$
where dA is the differential of area in whatever coordinate system you are using, in the xy-plane. Because of the circular symmetry I would recommend changing to polar coordinates- where the two coordinate variables can be separated.