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[Solved] Let D be a region bounded by a simple closed path C in the x-y plane. Use Green’s Theorem to prove that the coordinates of the centroid (x̄,ȳ)

Question: Let D be a region bounded by a simple closed path C in the x-y plane. Use Green’s Theorem to prove that the coordinates of the centroid \(\left( \bar{x},\bar{y} \right)\) of D are:

\[\bar{x}=\frac{1}{2A}\oint\limits_{C}{{{x}^{2}}dy}\] \[\bar{y}=\frac{1}{2A}\oint\limits_{C}{{{y}^{2}}dx}\]

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