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Math Help - Finding Polar Limits of Integration

  1. #1
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    Finding Polar Limits of Integration

    I'm currently reading an AP Calculus BC review book, and one of its example problems has to do with finding the area of the inner loop of a limašon.

    The formula for the limašon is r=1+2\cos\theta. I'm supposed to find the area of just its inner loop. The book begins its problem walkthrough by stating that "in the inner loop, r sweeps out the region as \theta goes from \frac{2\pi}{3} to \frac{4\pi}{3}." Those would, therefore, be the limits of integration.

    I'm really shaky on polar coordinate geometry, though, so I have no idea how you're supposed to have figured that out. How do you find the \theta values on a polar coordinate graph? I'm familiar with the basic polar-cartesian conversion equations x=r\cos\theta, y=r\sin\theta, r=\sqrt{x^2+y^2}, and \theta=\arctan{\frac{y}{x}}, but I don't know how to find \theta values when the polar graph is as complicated as a looped limašon.

    Thanks for any help you could give me!
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by greenstupor View Post
    I'm currently reading an AP Calculus BC review book, and one of its example problems has to do with finding the area of the inner loop of a limašon.

    The formula for the limašon is r=1+2\cos\theta. I'm supposed to find the area of just its inner loop. The book begins its problem walkthrough by stating that "in the inner loop, r sweeps out the region as \theta goes from \frac{2\pi}{3} to \frac{4\pi}{3}." Those would, therefore, be the limits of integration.

    I'm really shaky on polar coordinate geometry, though, so I have no idea how you're supposed to have figured that out. How do you find the \theta values on a polar coordinate graph? I'm familiar with the basic polar-cartesian conversion equations x=r\cos\theta, y=r\sin\theta, r=\sqrt{x^2+y^2}, and \theta=\arctan{\frac{y}{x}}, but I don't know how to find \theta values when the polar graph is as complicated as a looped limašon.

    Thanks for any help you could give me!
    To get to the inner loop of the limašon, the idea is to find values of \theta that cause r to be negative!

    So solve the inequality 1+2\cos\theta<0\implies\cos\theta<-\tfrac{1}{2}. It turns out in this case that \frac{2\pi}{3}< \theta<\frac{4\pi}{3} does the job.

    Does this help?
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  3. #3
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    Your answer metaphorically blew my mind.

    Thank you!
    Last edited by greenstupor; April 18th 2010 at 10:35 PM.
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