Limits of integration in area enclosed by polar curves

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?

• What values does $\theta$ have to take on in order to produce a "rose with three petals"? – Brian Jan 23 '20 at 3:44

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 $$2\pi$$.
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.