The cylinder is what is removed from the sphere.

Think of beads on a string.

The string is the cylinder.

I finally think I understand where my confusion is coming from on this. The total volume removed from the sphere=the volume of the cylinder+the volume of the endcaps, yes? And so, were one to drill a "six inch hole" in the earth, the actual CYLINDER would have a length of six inches, and the "drill bit" would still have to be long enough to go all the way through, leaving a six-inch-thick ring with an arc-shaped cross section. Am I getting it? I had been envisioning the result of a six-inch drill bit prior to this revelation. But if what I wrote above is correct (I think it must be), then to say "the cylinder is what is removed from the sphere" isn't completely true, because the end-caps are also removed.

I'm still impressed by your story, though.

If the length of the cylinder is "L," is the remaining volume piL^2?