# How do I solve this problem?

#### HallsofIvy

##### Elite Member
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 $$\displaystyle \frac{\pi}{6}$$.