Astronomers use a technique called stellar stereography to determine the density of stars in a star cluster from the observed (two-dimensional) density that can be analyzed from a photograph. Suppose that in a spherical cluster of radius $R$ the density of stars depends only on the distance $r$ from the center of the cluster. If the perceived star density is given by $y(s)$, where $s$ is the observed planar distance from the center of the cluster, and $x(r)$ is the actual density, it can be shown that $$y(s)=\int_{s}^{\,R} {\frac{2r}{\sqrt{r^2-s^2}}x(r)}\,dr.$$ If the actual density of stars in a cluster is $x(r)=\frac{1}{2}(R-r)^2,$ find the perceived density $y(s)$.

Note: Your answer could contain $R$ and $s$.

$y(s) =$