Area inside a curve and outside a Cardoid

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HINT...You need to establish where the curves intersect in order to determine the limits, so solve $3\cos \theta=1+\cos \theta$.

So your limits are $\pm\frac {\pi}{3}$

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james
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Updated on August 10, 2020

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  • james
    james over 3 years

    For a National Board Exam:

    Find the area which is inside the curve r=3cos(theta) and outside the cardoid r=1+cos(theta)

    Answer is pi

    Ok I am trying to setup the right definite integral for the calculator, I've tried:

    Using the formula for finding area of polar curves: $${ A = \int^b_a \frac{1}{2} f(\theta)^2 d\theta}$$

    $${ A = \int^{2\pi}_0 \frac{1}{2} ( 3cos(\theta) - (1+cos(\theta)))^2 d\theta = 3\pi}$$

    What is the right integral?

    • achille hui
      achille hui over 8 years
      1) There are typos in your integrand. 2) You need to integrate over the range where $3\cos\theta \ge \max( 1 + \cos\theta, 0 )$, not over $[0,2\pi)$.
  • james
    james over 8 years
    i use these as limits but i get 0.543... is it my calculator? i also try pi/3 to 5pi/3 and other iterations of the same thing...