For center of mass problems, it's possible to use a variable mass distribution. For a half cylinder, I've been looking at different mass distributions (constant, 1/r, 1/r^2, etc.) My teacher pointed out that at some point it runs into problems. I found that this was the case at 1/r^3, as you evaluate ln(r) from 0 to R, which gives infinite mass. Could someone please provide a physical explanation as to why this is so different from 1/r^2? (Which gives half the diameter times the charge density magnitude, by the way). Thanks.