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centroid of regions is same as average value of function

Source: Putnam 1984 B4

September 3, 2021
functionintegrationcalculus

Problem Statement

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