3D Change of Variables

SigurRos

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!

CarlB

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?

Carl

benorin

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

[tex]x^2(z^3+1)^2+y^2(z^3-1)^2=(z^6-1)^2[/tex]

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

[tex]u=x(z^3+1),v=y(z^3-1),w=z^6-1[/tex]

so that the equation then becomes

[tex]u^2+v^2=w^2[/tex]

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?