Is there a clever method for integrating over asymetric paraboloids

[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?

 

[lippyka]

Yes, there is a trick to it. The trick is to first consider the paraboloid's equation, z = f(x,y), and then use the equation for an implicit ellipse, Ax^2 + By^2 + Cxy + Dx + Ey + F = 0, to define the boundary of the elliptical cylinder. You can then use the equation of the paraboloid and the equation of the ellipse together to solve for the portion of the paraboloid enclosed within the elliptical cylinder. This is a relatively complex problem, but it is doable. The idea is to first solve the paraboloid equation for x in terms of y, and then substitute this into the ellipse equation to get a single variable equation. Once you have this single variable equation, you can then find the boundaries of the elliptical cylinder by solving it for y, and then use these boundaries to calculate the volume of the paraboloid enclosed in the cylinder.For example, if we consider the equation:z = 6 -7x^2 -y^2we can solve for x in terms of y as:x = ±√(6 − y²)/7Substituting this into the ellipse equation, we can get a single variable equation:Ay² + By² + D + E + F = 0Solving this equation for y will give us the boundaries of the elliptical cylinder. We can then use these boundaries to calculate the volume of the paraboloid enclosed in the cylinder.