 Center of mass of a cone with a power-law radial density

For a solid cone of base radius $a$, height $h$ and density $\rho (r)={\rho }_0(r/a{)}^p$, take $p>-2$ so the axis singularity, if present, is integrable. In cylindrical coordinates, at fixed radius $r$ the vertical range is $0\le z\le h(1-r/a)$. Integrating over height first givesThe first height moment is $\pi {\rho }_0{\int }_0^a(r/a{)}^{p}r{h}^2(1-r/a{)}^{2}dr=2\pi {\rho }_0{a}^2{h}^2/[(p+2)(p+3)(p+4)]$. Dividing proves $d_p=h/(p+4)$. Thus uniform density has centroid $h/4$, while density proportional to radius has centroid $h/5$. For sums of such density components, total masses and first moments add, giving a mass-weighted centroid.