Green's theorem question?

Apr 2010
487
9
The question
Suppose R is a closed region in the plane bounded by a closed non-self-intersecting, piecewise smooth plane curve \(\displaystyle \tau\). Prove that the area of R is given by

\(\displaystyle \frac{1}{2}\oint_{\tau}{x \ dy - y \ dx}\)

I'm unsure of how to solve this. Could someone guide me? Thanks.
 

TheEmptySet

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The question
Suppose R is a closed region in the plane bounded by a closed non-self-intersecting, piecewise smooth plane curve \(\displaystyle \tau\). Prove that the area of R is given by

\(\displaystyle \frac{1}{2}\oint_{\tau}{x \ dy - y \ dx}\)

I'm unsure of how to solve this. Could someone guide me? Thanks.
Just apply Green's theorem

\(\displaystyle \frac{1}{2}\oint_{\tau}{x \ dy - y \ dx} =\frac{1}{2} \iint_{D} 1+1 dA=\iint_D dA\)
 
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Apr 2010
487
9
Thanks, I didn't realise it was that simple. :)