How do I determine the area that is inside the circle r=3cos(θ) and outside the r=3sin(2θ) curve (For the first quadrant)

Question

I have this graph here, but how do I find out the area that's inside the circle but outside the 3sin2θ rose shaped curve? I'm lost, please help me solve this

Answer 1

The area of a region in polar coordinates defined by the equation $r=f(θ)$ with $\alpha\leq\theta\leq\beta$ is given by:$$\int_{\alpha}^{\beta} \frac{1}{2}{r}^2 \; d\theta$$

The area of a region between two functions in polar coordinates is just the difference in area between the two functions in that region . First, find $\alpha$ and $\beta$, which is where $r$ is equal in the two functions: $$3\cos{\theta}=3\sin{2\theta} \implies \theta=\pm\frac{\pi}{6}$$ Note that you're looking for the area only in the first quadrant. Try finishing the rest given what I have provided here.