Let $R$ be the region that lies between the curves $y=x^m$ and $y=x^n,\; 0 \le x \le 1,$ where $m$ and $n$ are integers with $0 \le n < m.$ Find the exact coordinates of the centroid of $R$ in terms of $m$ and $n$.

$\bar{x} =$

$\bar{y} =$

Note: Observe that the centroid of $R$ may not necessarily be in $R$ !