How do I solve this problem?

[math_N00b]

I have no idea where to begin or how do I go on about solving this, need desperate help. Thanks. You guys been absolute legends for helping me promptly for my previous 2 questions. God bless.

The question:

 

[HallsofIvy]

First, have you graphed it? If you don't mind "cheating" a little, you can use https://www.desmos.com/calculator.
Just enter \(\displaystyle r= 3 cos(\theta)\) for the first function and \(\displaystyle r= 3 sin(2\theta)\) for the second.

One thing you will need to know is where the two graphs intersect. That is, solve \(\displaystyle 3 cos(\theta)= 3 sin(2\theta)\). That is, of course, equivalent to \(\displaystyle cos(\theta)= sin(2\theta)\). There is a "double angle identity that says that \(\displaystyle sin(2\theta)= 2sin(\theta)cos(\theta)\) (that's probably in your text book) so the equation is \(\displaystyle cos(\theta)= 2 sin(\theta)cos(\theta)\). If \(\displaystyle cos(\theta)\ne 0\) (i.e. for \(\displaystyle \theta\ne \pi/2\)) we can divide both sides by \(\displaystyle cos(\theta)\) to get \(\displaystyle 2sin(\theta)= 1\), \(\displaystyle sin(\theta)= \frac{1}{2}\). That's true for \(\displaystyle \theta= \frac{\pi}{6}\) radians. You can see from the graph that this is symmetric about the x-axis so you want to integrate \(\displaystyle 3 cos(\theta)- 3 sin(2\theta)\) from \(\displaystyle -\frac{\pi}{6}\) to \(\displaystyle \frac{\pi}{6}\).