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Is there a clever method for integrating over asymetric paraboloids

  1. Aug 23, 2011 #1
    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?
     
  2. jcsd
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