Area inside a curve and outside a Cardoid
2,086
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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Author by
james
Updated on August 10, 2020Comments
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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?
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achille hui over 8 years1) 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 $[0,2\pi)$.
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james over 8 yearsi 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...