How do I calculate an area in Mathematica?

[IridescentRain]

Hi!

Say I have a region described by any number of inequalities. This region is a surface in 3D space. How can I ask Mathematica to calculate the region's area?

If it helps, my particular region is the intersection of a hollow sphere and a solid (i.e. filled-out) toroid-like surface. I'm using Mathematica 9, but I think I can find a Mathematica 8 .dmg file and my old license key if necessary.

Thanks!

 

[Bill Simpson]

Since no one has offered any idea and perhaps you might still see this...

Suppose you randomly generate a point on the surface of your sphere.
Then check to see whether that point is inside the toroid like surface.
Repeat that perhaps 10^6 times. See what fraction of those are inside.
Does that fraction tell you something interesting or useful?

Note: If you did not have a hollow sphere to start with then this might
be more complicated. You might think about why that might be.

 

[IridescentRain]

Thanks for the suggestion. With a million points, that would make a very good approximation. (Obviously, the area of my region would be the sphere's area times the fraction of points inside the torus.)

Cheers!

 

[Bill Simpson]

Wonderful. Just be very careful with that random generation so that the points are uniformly distributed across the sphere. Doing something like randomly choosing a latitude and then randomly choosing a longitude would not make the points uniformly distributed and would lead to an inaccurate estimate of the area.

 

[IridescentRain]

Ah, thanks for the advice. It's perfectly reasonable now that you've mentioned it, yet I would have probably gone and done just that --randomly select a latitude and a longitude-- without realising that the point distribution would then be denser near the poles. I'll be more careful than that with my random-point selection.

Cheers!