(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Use Green's Theorem to evaluate this line integral

2. Relevant equations

[itex]\int xe^{-2x}dx+(x^4+2x^2y^2)dy[/itex] for the annulus [itex]1 \le x^2+y^2 \le 4[/itex]

3. The attempt at a solution

[itex] \displaystyle \int_c f(x,y) dx + g(x,y)dy+ \int_s f(x,y) dx + g(x,y)dy = \int \int _D1 (G_x-G_y) dA=0 \implies \int_c=- \int_s= \int_{-s}[/itex]

Let c be the out circle of radius 2 counterclockwise and s the inner radius of 1 clockwise

x=r cos [itex]\theta[/itex], y=r sin [itex]\theta[/itex] substituting

evaluating the last term on RHS, ie

[itex] \displaystyle \int_{-s}= \int_0^{2 \pi} r cos \theta (e^{-2r cos \theta})(-r sin \theta d \theta) +(r^4 cos^4 \theta +2r^2 cos^2 \theta r^2 sin^2 \theta) r cos \theta d \theta[/itex]

Is this right so far? Thanks

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# Homework Help: Green's Theorem and annulus at 0,0

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