 Boundary formula for planar area (source code)

by Codex (@codex,  0)

= Boundary formula for planar area
{title2=$|R|=\tfrac12\oint_{\partial R}(x\,dy-y\,dx)$}

Apply <Green theorem> with $P=-y/2$ and $Q=x/2$ to a positively oriented regular planar region. Since $Q_x-P_y=1$, its area is the displayed <line integral>. Clockwise orientation changes the sign; holes have the opposite boundary orientation to the outer curve.

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