= Center of mass of a cone with a power-law radial density
{title2=$M_p=2\pi\rho_0a^2h/[(p+2)(p+3)],\quad d_p=h/(p+4)$}
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 $0\leq z\leq h(1-r/a)$. Integrating over height first gives
$$
M_p=2\pi\rho_0\int_0^a(r/a)^p r h(1-r/a)dr=\frac{2\pi\rho_0a^2h}{(p+2)(p+3)}.
$$
The first height moment is $\pi\rho_0\int_0^a(r/a)^p r h^2(1-r/a)^2dr=2\pi\rho_0a^2h^2/[(p+2)(p+3)(p+4)]$. 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.
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