 Power-law initial-mass-function tail coordinate

For an initial mass function proportional to $M^{-p}$ on $M\ge M_{\min }$ with $p>1$, integration and normalization give the displayed upper-tail fraction. The probability integral transform makes $X$ uniform on $[0,1]$ for a randomly sampled birth mass. Evolutionary lifetime regions can then be measured as ordinary areas when the independent age coordinate is also uniform.