- #1

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## Homework Statement

Let C be the line segment connecting the points (x

_{1},y

_{1}) and (x

_{2},y

_{2}). More over let the line integral over C of (x dy - y dx) = x

_{1}y

_{2}- x

_{2}y

_{1}.

Suppose the vertices of a polygon, listed in counter-clockwise order, are (x

_{1}y

_{1}), (x

_{2}y

_{2}), ... , (x

_{n}y

_{n}). Show that the area of the polygon is

(1/2) * ((x

_{1}y

_{2}- x

_{2}y

_{1}) + (x

_{2}y

_{3}- x

_{3}y

_{2}) + ... + (x

_{n}y

_{1}- x

_{1}y

_{n}))

## Homework Equations

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, (x

_{1}y

_{1}), (x

_{2}y

_{2}), ... , (x

_{n}y

_{n}). Then using the fact that the line integral over C of (x dy - y dx) = x

_{1}y

_{2}- x

_{2}y

_{1}, 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"