Let C be the line segment connecting the points (x1,y1) and (x2,y2). More over let the line integral over C of (x dy - y dx) = x1y2 - x2y1.
Suppose the vertices of a polygon, listed in counter-clockwise order, are (x1y1), (x2y2), ... , (xnyn). Show that the area of the polygon is
(1/2) * ((x1y2 - x2y1) + (x2y3 - x3y2) + ... + (xny1 - x1yn))
I don't know a relevant equation, but I suspect there probably is one.
The Attempt at a Solution
So, basically, I just want to say something like, let C* be the set of all line segments that connect, with positive orientation, (x1y1), (x2y2), ... , (xnyn). Then using the fact that the line integral over C of (x dy - y dx) = x1y2 - x2y1, and by repeatedly applying this fact, I would have something like:
[itex]\sum[/itex] * = 1 n of ([itex]\int[/itex]C* (x dy - y dx)). I think this gives the desired result except for the 1/2, which still eludes me.
Also, how do you format these sigmas? It's supposed to read "Sigma from *=1 to n"