Existence, Uniqueness & Bifurcations · Challenge problem

For each real $r$, the equation $y' = (y - r)(y - r^2)$ has the equilibria $y = r$ and $y = r^2$. For $r \ne 0, 1$ exactly one of them is stable; call it $s(r)$. Evaluate $\displaystyle\int_0^2 s(r)\,dr$.