Limits of integration in area enclosed by polar curves

Question

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?

Answer 1

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.

Answer 2

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 $2\pi$.