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I'm attempting to use Green's Theorem to express the area of a region in the complex plane in terms of a contour integral, but I'm a little confused as to how this works. I have a simple closed curve $\gamma$ with interior $D$, and I believe I'm supposed to get $\mathrm{Area}(D)=\frac{1}{2i} \oint_\gamma \overline{z} \,dz.$ Can anyone help me justify this?

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$\int_\gamma \bar{z}dz = \int_\gamma (x - iy)(dx + idy) = \int_\gamma (xdx + ydy) + i \int_\gamma( xdy - ydx)$now hit this with stokes:$ = \int_D d(xdx + ydy) + i\int_D d(xdy - ydx) = i \int_D 2 dx \wedge dy$

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    Because the first integral is zero, you can also multiply it by $i$ and add it on to the second one to justify $ i \int 2 x dx = 2 i \text{Area(D)}$ i.e. $\int x dx = \text{Area(D)}$2014-07-20