[SOLVED] Check out my result : line integral

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June 14th 2009, 07:53 PM
arbolis

[SOLVED] Check out my result : line integral

$I=\int_C \frac{dx+dy}{x^2+y^2}$ where $C$ is described by $r(t)=(\cos t, \sin t)$ , $0\leq x \leq 2\pi$ .
My attempt : $|r'(t)|=1$ so $I=\int_0^{2\pi} \frac{\cos t dt+\sin tdt}{\cos^2t + \sin^2t}=0$ .
I realize that $C$ is a circle of radius $1$ and that if my result is good then $F(x,y)=\frac{dx+dy}{x^2+y^2}$ is a conservative field. But I'm not sure I've done it right.
June 14th 2009, 07:58 PM
Jhevon

Quote:

Originally Posted by arbolis

$I=\int_C \frac{dx+dy}{x^2+y^2}$ where $C$ is described by $r(t)=(\cos t, \sin t)$ , $0\leq x \leq 2\pi$ .
My attempt : $|r'(t)|=1$ so $I=\int_0^{2\pi} \frac{\cos t dt+\sin tdt}{\cos^2t + \sin^2t}=0$ .
I realize that $C$ is a circle of radius $1$ and that if my result is good then $F(x,y)=\frac{dx+dy}{x^2+y^2}$ is a conservative field. But I'm not sure I've done it right.

you didn't do it right.

here we have $x = \cos t \implies \boxed{dx = - \sin t~dt}$ and

$y = \sin t \implies \boxed{dy = \cos t~dt}$

(the answer is still 0 though :))
June 14th 2009, 08:04 PM
arbolis

Thanks for the correction. I do still get 0 as you pointed out.
June 14th 2009, 08:16 PM
Jhevon

Quote:

Originally Posted by arbolis

Thanks for the correction. I do still get 0 as you pointed out.

indeed. by now you should have it memorized that integrating sine or cosine over any multiple of their period yields zero. so for instances, $\int_0^{2 \pi} \sin t~dt = \int_\pi^{3 \pi} \sin t~dt = \int_{\pi /2}^{5 \pi / 2} \cos t~dt = \int_\pi^{9 \pi} \sin t~dt$ $= \int_a^{a + 2k \pi} \sin t ~dt = \int_k^{2n \pi} \cos t~dt = 0$ etc etc etc