Disclaimer: use at your own risk. Its probably wrong because I don't remember all the formulas (the ones below I've looked up, but I didn't find anything for "partial spheres").

First here's what the problem says:

0a) x^2 + y^2 >= 0
0b) x^2 + y^2 <= 4
0c) d(x,y,z) = SQRT ( 9 - x^2 - y ^2) = SQRT ( 9 - (x^2 + y^2) )
0d) z = d(x,y,z)

Then here's some formulas courtesy of wikipedia:

1. volume = mass / average density
2. volume(sphere) = 4/3*Pi*radius^3
3. radius^2 = x^2 + y^2 + z^2

Then 3)
radius^2 = x^2 + y^2 + 9 - (x^2 + y^2) (replace z, sqrt goes away)
radius^2 = 9
radius = 3

And 2)
volume = 4/3*Pi*3^3 = 4*Pi*3^2 = 36*Pi

Thus 1)
mass = volume * average density = 36*Pi * average density

And here's the genius part:
average density = (minimum density + max density)/2

Since d = f(-(x^2+y^2)) and (x^2 + y^2) always grows when x and y grows, min(d) is when x^2 + y^2 = 4 and max(d) when = 0.
So min(d) = SQRT(9 - 4) and max(d) = SQRT (9-0)
Or min(d) = 2.2361 and max(d) = 3

Therefore average density = (2.2361 + 3)/2 = 2.6181

So mass = 36*Pi * 2.6181 = 296.09

There's a catch: the volume formula is probably wrong because I've done it for a full sphere. How much of a "partial sphere" are we talking about? If it's half, divide the mass by 2.

Also, we're talking about a hollow sphere, so I don't know how that affects the formulas. Maybe it doesn't because the impact may be reflected in the density formula.







[download a life]