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I need to compute the line integral for the vector $\vec{F} = \langle x^2,xy\rangle$, for the curve specified: part of circle $x^2+y^2=9$ with $x \le0,y \ge 0$,oriented clockwise.

Once again, I'm stuck at the setup (this happens a lot with me). I know that I need to parameterize F, but how would I go about doing this exactly?

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oops, I fixed it. –  duxrule May 22 '13 at 11:37

2 Answers 2

up vote 3 down vote accepted

We have $$F(x,y)=x^2\textbf{i}+xy\textbf{j}$$ and $$C: x^2+y^2=1, x\le0, y\ge0$$ and want to evaluate $$\oint_CF\cdot dr=\int_{\pi}^{\pi/2} F(\cos t,\sin t)\cdot(-\sin t,\cos t)dt=\int_{\pi}^{\pi/2}(\cos^2 t,\sin t\cos t)\cdot(-\sin t,\cos t)dt\\\\\\ =\int_{\pi}^{\pi/2}(-\sin t\cos^2 t+\sin t\cos^2 t)dt=0$$

enter image description here

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definitely enough. thanks! –  duxrule May 22 '13 at 11:59
    
+1 for the graph. What program did you make it in? –  Ataraxia May 22 '13 at 12:01
    
ah. Unfortunately no, but I had a professor who used it for lectures, so I'm familiar with it. Is it free/open source? –  Ataraxia May 22 '13 at 12:08
    
@ZettaSuro: Of course not. But if you get one, it'll be very helpful. :) –  B. S. May 22 '13 at 12:09
    
Great! Also, one thing I just realized: is the loop around the integral appropriate? This isn't a closed loop integral. –  Ataraxia May 22 '13 at 12:11

Another method is to use Green's theorem:

$$\oint_C \vec{F}\cdot d\vec{r}=\iint_D|\nabla\times \vec{F}|dA$$

Where D is the area within the loop.

$$|\nabla\times \vec{F}|=y - 2x$$

Convert to cylindrical coordinates:

$$y-2x = r\sin{\theta}-2r\cos{\theta}$$

And evaluate the following double integral:

$$\int_\pi^{\frac{\pi}{2}}\int_0^3r(\sin{\theta}-2\cos{\theta})r\space dr \space d\theta$$

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