Putnam 1984 B4

Find, with proof, all real-valued functions $y=g(x)$ defined and continuous on $[0,\infty)$ , positive on $(0,\infty)$ , such that for all $x>0$ the $y$ -coordinate of the centroid of the region $$R_x=\{(s,t)\mid0\le s\le x,\enspace0\le t\le g(s)\}$$ is the same as the average value of $g$ on $[0,x]$ .