Why does a 1/r³ mass distribution give infinite mass for a half cylinder?

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M-Speezy
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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.
 
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You did the maths - didn't you notice the difference during the calculation?
Your teacher pointed out that "it" runs into problems... what is "it"?
 
The math shows the theoretical distributions of mass, not necessary to be physically possible for any power of 1/r. So it's really a matter of discussing what actual densities are possible. You also asked to give a physical explanation to a seemingly math confusion, so you may want to specify what your asking.