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Problem 47 Hard Difficulty

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} = (\frac{1}{L}) \int x\ ds $ and $ \bar{y} = (\frac{1}{L}) \int 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 $.

Answer

$\bar{x}=8 / \pi, \bar{y}=3 / \pi$

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Video Transcript

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