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Math Help - Area of a cardioid

  1. #1
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    Area of a cardioid

    Hi all, can't quite figure this one out:
    I need to find the area inside the cardioid r = 1 + cos(theta)
    I set up an iterated integral, the outer of which is for theta from 0 to 2pi, and the inner is from 0 to 1+cos(theta) with respect to r, and the function is just f=1
    This gives me 2pi, but the solution is supposed to be 3pi/2. Can anyone point me in the right direction on how to calculate this? It would be greatly appreciated!
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  2. #2
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    I would use 2 integrals and each integral multiplied by 2. The first integral would be for the the piece in the 2nd quadrant and the piece in the 1st.
    Since you will multiple them by 2, you won't have to worry about the other sections.
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  3. #3
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    Just tried that...ended up with 2pi again :/
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  4. #4
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    Can you post your two integrals and bounds?
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  5. #5
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    Sure thing. Sorry I don't know how to code the text to make it look clean
    I had 2*(1+cos(theta)) from pi/2 to pi, and 2*(1+cos(theta)) from 0 to pi/2, evaluated i got 2pi - pi - 2 + pi + 2 = 2pi
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  6. #6
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    \displaystyle \mbox{Area of Cardioid}=\frac{1}{2}\int_0^{2\pi}r^2 d\theta
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  7. #7
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    Thanks!
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  8. #8
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    I was sick of trying to figure it out so I just looked in a book.
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  9. #9
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    Haha, I don't blame you :P Thanks for looking that up though, I couldn't find it!
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