The integral ${\int }_{\mathrm{A}}{\mathrm{\rho }}^2 \mathrm{dA}$ is given a special name — the polar moment of inertia $\mathrm{J}$:

$\mathrm{J}={\int }_{\mathrm{A}}{\mathrm{\rho }}^2 \mathrm{dA}$

and the relationship between the torque $\mathrm{T}$ transmitted by the shaft and the shear stress can be restated as:

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