# Is there a clever method for integrating over asymetric paraboloids

1. Aug 23, 2011

### Daron

There are a few questions on the forum about calculating the volume enclosed between an upwards-opening and a downwards-opening paraboloids, and I think I understand the method there. However they all involve symmetric paraboloids, and the intersection of the pair is always contained within a circular cylinder.

I tried to apply the method to a similar question with one symmetric and one asymmetric paraboloid, specifically

z = 6 -7x^2 -y^2

And here it is impossible to reduce to polar coordinates. So finding the volume of the paraboloid contained in a relevant elliptical cylinder is harder.

But then I remembered that there is no way to find the circumference of a non-circular ellipse. So I'm not sure if I can find a finite expression for the portion of the asymetric paraboloid contained within the cylinder that contains all intersections of the two paraboloids.

In short, how would I approach this sort of problem? Is there a trick to it?