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I am learning about finding the area enclosed by polar curves. I don't understand how to find the limits of integration to use. I know the formula is $\frac12\int_a^b f(\theta)^2\operatorname d\theta$, but how do you find the $a$ and $b$?

For example:

Find the area enclosed by $r=\cos(3\theta)$

I know this is a rose with three petals, but how do I figure out what to set $a$ and $b$ as?

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    $\begingroup$ What values does $\theta$ have to take on in order to produce a "rose with three petals"? $\endgroup$ – Brian Jan 23 '20 at 3:44
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When you have $r=\cos(k\theta)\,, k$ odd, there will be $k$ petals, traced out once as $\theta$ goes from $0$ to $\pi$.

A little trial and error will show you this. For $k$ even you will have $2k$ petals, traced out once as $\theta$ goes from $0$ to $2\pi$.

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Find the area A of the petal symmetric about the x-axis by taking $a=-\pi/6, b=\pi/6.$ Then the required area is 3A.

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