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Math Help - using polar coordinates to find area of cardioid

  1. #1
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    using polar coordinates to find area of cardioid

    Use polar coordinates to calculate \int \int_R \frac{1}{\sqrt{x^2+y^2}}dAwhere R is the region inside thecardioid r = 1 + sinq and above the x-axis.
    In polar form, the equation is just 1/r, right?
    I'm having a problem working out the limits here. For the inner integral are the limits 0\leq r \leq (1+sin\theta).
    What are the limits for the outer integral? Am i right in thinking the lower limit is also 0 as it's above the x-axis? What about the upper limit? Because it's a cardioid is it 2\pi?
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  2. #2
    MHF Contributor matheagle's Avatar
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    yes r=\sqrt{x^2+y^2} but don't forget the Jacobian.

    you also have dxdy=rdrd\theta

    above the x-axis means 0<\theta <\pi

    \int_0^{\pi}\int_0^{1+\sin\theta}drd\theta
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  3. #3
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    D'oh! I always forget about that r! I was hopelessly stuck because I was doing 1/r which of course integrates to log(r) & log(0) is undefined. Hence my total loss at what to do.
    That makes it a lot easier to do. Thanks for that.
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