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3D Change of Variables

  1. May 15, 2006 #1
    Our math professor gave us this take-home project:

    Consider a solid in the shape of the region D inside the surface

    x^2 / (z^3 - 1)^2 + y^2 / (z^3 + 1)^2 = 1

    If the density of the solid at the point (x,y,z) is x^2 + y^2 + z^2 then determine the mass of this solid. A GOOD CHANGE OF VARIABLES WILL HELP.

    I understand how to do the problem but I can't get a change of variables that works well. I've tried cylindrical and spherical and many other random ones. Can anyone suggest a good change of variables to use? My teacher said that the cross sections for integration are in the shape of ellipses. Thank you!
    Last edited: May 15, 2006
  2. jcsd
  3. May 16, 2006 #2


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    In problems which are of the form u^2 + v^2 = 1, you will frequently find that a useful change of variable is something like u = r\cos(\theta) and v= r\sin(\theta). Did you try this?

  4. May 17, 2006 #3


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    If you cross-multiply the given equation, you arrive at


    and so it would seem that a likely useful change of variables would be:


    so that the equation then becomes


    which is a cone in uvw-space; but I haven't figured out just what the solid in xyz-space is: what is that "region D" that you were given?
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