What Is the Best Change of Variables for Integrating a Complex 3D Solid?

[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

$$x^2(z^3+1)^2+y^2(z^3-1)^2=(z^6-1)^2$$

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

$$u=x(z^3+1),v=y(z^3-1),w=z^6-1$$

so that the equation then becomes

$$u^2+v^2=w^2$$

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?