Area of a Polygon Green's Theorem leads to a convenient formula for the area of a polygon.

(a) Let \(C\) be the line segment joining \(\left(x_{1}, y_{1}ight)\) to \(\left(x_{2}, y_{2}ight)\). Show that
\[
\frac{1}{2} \int_{C}-y d x+x d y=\frac{1}{2}\left(x_{1} y_{2}-x_{2} y_{1}ight)
\]
(b) Prove that the area of the polygon with vertices \(\left(x_{1}, y_{1}ight),\left(x_{2}, y_{2}ight), \ldots,\left(x_{n}, y_{n}ight)\) is equal [where we set \(\left.\left(x_{n+1}, y_{n+1}ight)=\left(x_{1}, y_{1}ight)ight]\) to
\[
\frac{1}{2} \sum_{i=1}^{n}\left(x_{i} y_{i+1}-x_{i+1} y_{i}ight)
\]

Transcribed Image Text:

THEOREM 1 Green's Theorem Let D be a domain whose boundary 3D is a simple closed curve, oriented counterclockwise. If F and F2 have continuous partial deriva- tives in an open region containing D, then aD aF2 Fi y = ff (3F/ - 3F) dA y F dx + F dy = 2