The Centroid via Boundary Measurements The centroid (see Section 15.5) of a domain \(\mathcal{D}\) enclosed by a simple closed curve \(C\) is the point with coordinates \((\bar{x}, \bar{y})=\left(M_{y} / M, M_{x} / Might)\), where \(M\) is the area of \(\mathcal{D}\) and the moments are defined by

\[
M_{x}=\iint_{\mathcal{D}} y d A, \quad M_{y}=\iint_{\mathcal{D}} x d A
\]

Show that \(M_{x}=\oint_{C} x y d y\). Find a similar expression for \(M_{y}\).