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I am required to prove that $\displaystyle \int_0^{2\pi} |x \cos(\theta)+y \sin(\theta)|\, d\theta= 4\sqrt{x^2+y^2}$, $\ x$ and $y$ are real.

I let $\sin\theta = \frac yz$, $\cos\theta=\frac xz$, where $z$ is supposedly complex. Then i managed to show that $x\cos\theta + y\sin\theta = z$, so i am left with integrating $|z|$ (which is the area of a circle?)

I am stuck here since the RHS of what i am supposed to prove, doesn't have pi inside.

Please advise, thanks!!

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0 = theta! thanks! –  PongYoPongYo Aug 20 '12 at 9:32
    
I edited in Latex and replaced $x$ by $z$ in the denominator of your $\cos\theta$ expression, is this right ? ($x/x$ looked suspicious...) –  Raymond Manzoni Aug 20 '12 at 9:43
    
Hi Raymond, yes it is. Thank you so much! –  PongYoPongYo Aug 20 '12 at 10:18
    
you are Welcome ! –  Raymond Manzoni Aug 20 '12 at 11:11
    
Satisfied with an answer below? –  Did Sep 19 '12 at 17:29
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3 Answers 3

Since $x=r\cos\alpha$ and $y=r\sin\alpha$ for some $\alpha$ with $r=\sqrt{x^2+y^2}$, the integral is $$ r\int_0^{2\pi}|\cos(\theta-\alpha)|\mathrm d\theta=r\int_0^{2\pi}|\cos(\theta)|\mathrm d\theta=4r\int_0^{\pi/2}\cos(\theta)\mathrm d\theta=4r\,\left[\sin(\theta)\right]_0^{\pi/2}=4r. $$

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Note that $x$ and $y$ are fixed real numbers.

The way to do this is to set $x=r\sin\phi$, $y=r\cos\phi$ so that $x^2+y^2=r^2$ and $\phi = \arctan {\frac x y}$.

Then the integrand becomes $\left|r\sin (\theta + \phi)\right|$. The factor $r$ can be extracted, and since the integral is around the unit circle, you need double the integral for the interval over which $sin (\theta + \phi) \geq 0$.

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Put $x=r\cos t$ and $y=r\sin t=>r=\sqrt{x^2+y^2}$ and $t=tan^{-1}\frac{y}{x}$

$\displaystyle \int_0^{2\pi} |x \cos(\theta)+y \sin(\theta)|\, d\theta$

$=\sqrt{x^2+y^2}\displaystyle \int_0^{2\pi} | \cos(\theta -t)|\, d\theta$ assuming x,y are independent of $\theta$

Now $\cos x<0$ for $\frac{\pi}{2}<x<\frac{3\pi}{2}$ and $\cos x≥0$, elsewhere

As the given range is 0 to $2\pi$ and $\cos(2\pi s+x)=\cos x$ for integral s, we can safely break the definite integral into the 4 quadrants i.e.,

$\displaystyle \int_0^{2\pi} | \cos(\theta -t)|\, d\theta$

$=\displaystyle \int_t^{\frac{\pi}{2}+t} \cos(\theta -t)\, d\theta$

$+\displaystyle \int_{\frac{\pi}{2}+t}^{\pi+t} (-\cos(\theta -t))\, d\theta$

$+\displaystyle \int_{\pi+t}^{\frac{3\pi}{2}+t}(-\cos(\theta -t))\, d\theta$

$+\displaystyle \int_{\frac{3\pi}{2}+t}^{2\pi+t} \cos(\theta -t)\, d\theta$

$=\sin(\theta-t)|_t^{\frac{\pi}{2}+t}$

$-\sin(\theta-t)|_{\frac{\pi}{2}+t}^{\pi+t}$

$-\sin(\theta-t)|_{\pi+t}^{\frac{3\pi}{2}+t}$

$+\sin(\theta-t)|_{\frac{3\pi}{2}+t}^{2\pi+t}$

$=(1-0)-(0-1)-(0-1)+(1-0)=4$

$\displaystyle \int_0^{2\pi} |x \cos(\theta)+y \sin(\theta)|\, d\theta=4\sqrt{x^2+y^2}$

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Nice solution lab bhattacharjee....... –  juantheron Dec 15 '13 at 7:12
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