Chapter 8.3, Problem 47E

The centroid of a curve can be found by a process similar to the one we used for finding the centroid of a region. If C is a curve with length L, then the centroid is ( $\bar{x}$, $\bar{y}$) where $\bar{x}$ = (l/L) ∫ x ds and $\bar{y}$ = (l/L) ∫ y ds. Here we assign appropriate limits of integration, and ds is as defined in Sections 8.1 and 8.2. (The centroid often doesn’t lie on the curve itself. If the curve were made of wire and placed on a weightless board, the centroid would be the balance point on the board.) Find the centroid of the quarter-circle $y=\sqrt{16-x^2},$ $0\le x\le 4$.