Let $D$ be a region with density $1$ bounded by a simple closed path $C$ in the $xy-$plane. Use Green's Theorem to prove that the coordinates of the centroid $(\stackrel{}{x}\frac{,}{b}ar(y))$ are

$\stackrel{}{x}=\frac{1}{2A}o{\displaystyle \underset{C}{\overset{\phantom{}}{}}}{x}^{2}dy\frac{,}{b}ar(y)=-\frac{1}{2A}o{\displaystyle \underset{C}{\overset{\phantom{}}{}}}{y}^{2}dx$

where $A$ is the area of $D.$