Attachment 27490

How would I use Green's Theorem to prove this area integral?

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- Mar 11th 2013, 07:11 PMsdsu619Greens Theorem
Attachment 27490

How would I use Green's Theorem to prove this area integral? - Mar 11th 2013, 07:43 PMmajaminRe: Greens Theorem
It is a known deduction of Green's Theorem that, for an area of a disc in a plane,

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

Find an expression of x and y in terms of $\displaystyle r=f(\theta)$. and substitute into this equation. The only problem with this method is that this does not use Green's theorem directly, only a major consequence of it. but is probably still valid.

For $\displaystyle r=3\sin{2\theta}$,

$\displaystyle \displaystyle A = \frac{1}{2}\int_0^{\pi/2} r^2 d\theta = \frac{1}{2}\int_0^{\pi/2} (3\sin{2\theta})^2 d\theta = \frac{9}{2}\int_0^{\pi/2}\sin^2 2\theta d\theta = \frac{9}{2}\int_0^{\pi/2}\frac{1-cos4\theta}{2}d\theta =$