$\int \mathrm{dM}={\int }_{\mathrm{A}}\frac{{\mathrm{\tau }}_{\max }}{\mathrm{c}}{\mathrm{\rho }}^2 \mathrm{dA}$

${\mathrm{\tau }}_{\max }$ is the shear stress on the outer surface of the shaft and $\mathrm{c}$ is the radius of the shaft. These values are constant within area $\mathrm{A}$.
(No matter which area $\mathrm{dA}$ we consider, ${\mathrm{\tau }}_{\max }$ is always the shear stress on the outer surface and $\mathrm{c}$ is always the radius.)