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# Green's Theorem and Area

Show that the area bounded by a simple closed curve is given by $$\frac{1}{2} \oint{xdy-ydx}$$.

Solution

We use Green's Theorem, and let $$P=-y$$ , $$Q= x$$. Thus

$\oint { Pdx+Qdy } =\iint { \left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y} \right) dxdy }$

$\oint { -ydx+xdy } =\iint { \left(\frac{\partial (x)}{\partial x}-\frac{\partial (-y)}{\partial y} \right)dxdy }$

$\oint { xdy-ydx } = 2\iint {dxdy }$

$\oint { xdy-ydx } = 2\iint {A }$

$A = \frac{1}{2}\oint { xdy-ydx }$

Check out my other notes at Proof, Disproof, and Derivation

Note by Steven Zheng
2 years, 10 months ago

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This very question was asked in my semester examination which I could happily answer using the process described here.

- 2 years, 10 months ago