Sorry I took so long again. I spent a day or so trying to find a geometry similar to that of a disk again which would just integrate into a single formula without running it over points, but still no luck so far. I also started to draw up a graph for the rotation speeds for a uniform disk as compared to a sphere, but it wasn't very interesting. The rotation speeds for a sphere just started off at (0,0) and ran in a straight line to (1,1), while that for a disk starts off in the same way, following the same line, then begins to deviate as it approaches the rim, until it curves up faster toward the rim to about twice the rotation speed as that of a sphere for the thickness of a disk we've been finding for. I might still be able to identify the formula for that curve to find a formula for a disk, though, if it turns out to be a hyperbola or something, but for now, I just want to get the iteration for the reverse problem going.

I tried that last formula you posted, and this time I got about 4 times the rotation speed for the first ring, so it's getting closer, but I don't trust the formula because I don't know where it came from, except as a clever rig to fit the rotation curve, which it might come close to doing over all of the rings, but I can't be sure about the effects of the individual rings on particular points, which I'll need for the reverse problem. Besides that, I still wanted something just a little more precise, so I went ahead and broke up that "last integration over x" program for a disk I had into individual rings. I then made an array for the acceleration of gravity for each ring as they would act on a particle placed outside of every ring as (i,j), where i denotes the ring and j the position of the particle. It builds the array in about 5 seconds for ten rings, with about 99.99965% accuracy at the inner ring and 99.987% at the rim, so it is still very precise also.

It needed to build the array only once, and I could then find for the acceleration and rotation speeds for any of the ten positions by simply having the program add up the accelerations for each of the ten rings for any particular position, using the array, which only takes a fraction of a second. I can also have the program go through the array over and over for the different positions, while giving each of the rings different densities, also in a fraction of a second, using the same array in combination with another array for the densities of the rings, and find for the rotations speeds over various density configurations. That is what I will use for the reverse problem. I will still be crossing my fingers that it will eventually level off at the corresponding rotation speeds and densities for the rings the way I'm hoping it will, though. If not, I will need to try other methods for the reverse problem as well.

__________________
Let's put together the pieces of The Grand Puzzle . (website)

"Let's define another operator, Sz, which we won't pay any attention to."
"This transformation will automatically make zero equal zero."
"It may be true that zero equals zero -- and that is certainly an equality -- but I don't want to go into the details at this time."