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Math Help - Green's Theorem

  1. #1
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    Green's Theorem

    Hey,

    I have a tutorial question that is starting to bug me.
    I know the formula is:

    A = \frac{1}{2}\oint_{C}(x\,\mathrm{d}y + y\, \mathrm{d}x)
    I have the problem:
    Using Green's theorem, find the area of the disc of radius a.

    I have notes and a previous example, but unsure where to start, Do I use the fact that:
    x^2+y^2=r^2 where r =1

    cheers
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  2. #2
    Behold, the power of SARDINES!
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    Quote Originally Posted by ramdrop View Post
    Hey,

    I have a tutorial question that is starting to bug me.
    I know the formula is:

    A = \frac{1}{2}\oint_{C}(x\,\mathrm{d}y + y\, \mathrm{d}x)
    I have the problem:
    Using Green's theorem, find the area of the disc of radius a.

    I have notes and a previous example, but unsure where to start, Do I use the fact that:
    x^2+y^2=r^2 where r =1

    cheers
    In polar coordinates with r=1 you get

    x=\cos(\theta) \implies dx=-\sin(\theta)d\theta and
    y=\sin(\theta) \implies dy=\cos(\theta)d\theta

    Now \theta=0..2\pi
    Can you finish from here?
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  3. #3
    MHF Contributor FernandoRevilla's Avatar
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    Use the fact that:

    C \equiv\begin{Bmatrix} x=a\cos t\\y=a\sin t\end{matrix}\quad(t\in [0,2\pi])


    Fernando Revilla

    Edited: Sorry, I did't see TheEmptySet's post.
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  4. #4
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    ah okay, I was right in assuming that, cheers guys
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  5. #5
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    I got a final answer of 0..

    having replacing:
    x = cos{theta}, y = sin(theta), x1 = -sin(theta), y1 = cos(theta)

    then subbing it in I got:
    cos^2(theta) - sin^2(theta) with the limits 2pi and 0.

    I then changed this to cos(2*theta) by the formula and integrated to get:
    0.25sin(2*theta)

    and then finally putting the limits in, i got 0, surely it shouldnt be 0..
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  6. #6
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
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    Quote Originally Posted by ramdrop View Post
    I got a final answer of 0..

    having replacing:
    x = cos{theta}, y = sin(theta), x1 = -sin(theta), y1 = cos(theta)

    then subbing it in I got:
    cos^2(theta) - sin^2(theta) with the limits 2pi and 0.

    I then changed this to cos(2*theta) by the formula and integrated to get:
    0.25sin(2*theta)

    and then finally putting the limits in, i got 0, surely it shouldnt be 0..
    I didn't notice at first but you have a typo in your formula it should be

    \displaystyle \frac{1}{2}\oint xdy-ydx

    You need to change the sign on the sine term (haha pun intended!) then you will get the answer you wish
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  7. #7
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    So stupid, i copied it from my notes and i must have wrote it down wrong, because i haad both a + and a -, so I took a guess... my bad

    I got pi now
    Last edited by ramdrop; February 2nd 2011 at 01:47 PM.
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