Consider a solid sphere of radius R and mass density ρ(r) = ρ _ 0...

The density of any point inside the sphere is

$ρ = {ρ}_{0}\left(1 - \dfrac{{r}^{2}}{{R}^{2}}\right)0 < r \le R$

weight = buoyant force
mg = B

$mg = {ρ}_{L}Vg$
$∫ ρ\left(4\pi{r}^{2}dr\right)g = {ρ}_{L}\dfrac{4}{3}\pi{R}^{3}g$
$\mathop{∫}_{0}^{R}{ρ}_{0}\left(1 - \dfrac{{r}^{2}}{{R}^{2}}\right)4\pi{r}^{2}dr = {ρ}_{L}\dfrac{4}{3}\pi{R}^{3}$
$\mathop{∫}_{0}^{R}{ρ}_{0}4\pi\left({r}^{2} - \dfrac{{r}^{4}}{{R}^{2}}\right)dr = {ρ}_{0}4\pi{\left(\dfrac{{r}^{3}}{3} - \dfrac{{r}^{5}}{5{R}^{2}}\right)}_{0}^{R} = {ρ}_{L}\dfrac{4}{3}\pi{R}^{3}$
$\dfrac{2}{5}{ρ}_{0} = {ρ}_{L}$