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Homework Help: Is this integral set up correctly?

  1. Nov 24, 2006 #1

    G01

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    Find the area of the region within both circles;

    r=cos(theta) and
    r=sin(theta)

    using a double integral.

    I made the following:

    [tex] 2\int^{\pi/2}_{\pi/4} \int^{\cos\theta}_0 r dr d\theta [/tex]

    I multiplied by 2 because the area I have is only half of the total area to be found. Is this correct or am i doing something stupid?
     
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  3. Nov 24, 2006 #2

    NateTG

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    I'm not sure that it's a particularly good example for using multiple integrals, but you're effectively using Green's Theorem on something that can be readily handled by:
    [tex]A = \frac{1}{2} \int r^2 d\theta[/tex]
    or, for those who prefer not to deal with polar integrals:
    [tex]A=2 \int_0^{\frac{1}{2}} \left(\sqrt{(\frac{1}{2})^2-(x-\frac{1}{2})^2}-x \right)dx[/tex]

    I'm fairily sure that your expression will give a numerically correct result.

    http://mathworld.wolfram.com/GreensTheorem.html
     
    Last edited: Nov 24, 2006
  4. Nov 24, 2006 #3

    G01

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    Thanks Nate, I agree that its probably easier to do this with
    [tex]1/2 \int r^2 d\theta [/tex] but our assignment was to do it using a double integral. Thanks for the help. I'll keep working on it and see if i get the right answer.
     
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