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Math Help - Using a double integral to find the area of a circle.

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    Using a double integral to find the area of a circle.

    I have scanned the question I am working on and attached it as an image to this thread. A hint on this question would be good enough.
    Attached Thumbnails Attached Thumbnails Using a double integral to find the area of a circle.-001.jpg  
    Last edited by mr fantastic; January 28th 2011 at 05:54 PM. Reason: Re-titled.
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    Quote Originally Posted by Undefdisfigure View Post
    I have scanned the question I am working on and attached it as an image to this thread. A hint on this question would be good enough.
    Is this a unit circle?
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    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by Undefdisfigure View Post
    I have scanned the question I am working on and attached it as an image to this thread. A hint on this question would be good enough.
    Note that r=\cos\theta is a circle with center at \left(\frac{1}{2},0) and radius r=\frac{1}{2}.

    Similarly, r=\sin\theta is a circle with center at \left(0,\frac{1}{2}) and radius r=\frac{1}{2}.

    Now these circles will overlap at some point. So my hint is this:

    When considering the region of intersection, split it up into two separate regions. Due to how we find limits of integration in polar coordinates, each region will be defined over different ranges for \theta. So your integrals will look something like:

    \displaystyle\int_{\theta_0}^{\theta_1} \int_0^{\cos\theta} r\,dr\,d\theta+\int_{\theta_0^{\prime}}^{\theta_1^  {\prime}}\int_0^{\sin\theta}r\,dr\,d\theta

    where \theta_0\leq \theta\leq\theta_1 is the range you consider for the r=\cos\theta part, and \theta_0^{\prime}\leq \theta\leq\theta_1^{\prime} is the range you consider for the r=\sin\theta part.

    Can you proceed? If you still have issues, post back!
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    Oops, I didn't see somebody else posted.
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    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by Undefdisfigure View Post
    Yeah but what kind of limits of integration would I use?
    To figure out the limits of integration for \theta, its best to look at a graph showing the intersection of the two circles. The only additional hint I'll supply for now is that \theta=\frac{\pi}{4} is a value worth considering.
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