Area between two polar curves method

Question

The question is not too hard. I sketched them and they were correct which was not too bad.

I then did the second part by finding the intersection points between the two curves which are $\frac{\pi}{3}$ and $\frac{5\pi}{3}$.

I then integrated:

$$\int_{a}^{b} (r_o^2 - r_i^2)d\theta$$

$$\int_{\frac{\pi}{3}}^{\frac{5\pi}{3}} \frac{1}{2}((1-cos(\theta))^2 - (cos(\theta))^2) d\theta$$ and got $\frac{2\pi}{3}$+$\sqrt{3}$ as my answer.

Is my work (and hopefully my answer...) right or did I do something wrong?

Answer 1

You can use

$\displaystyle A=\int_{-\pi/3}^{\pi/3}\frac{1}{2}\left((\cos\theta)^2-(1-\cos\theta)^2\right)d\theta=2\int_0^{\pi/3}\frac{1}{2}\left((\cos\theta)^2-(1-\cos\theta)^2\right)d\theta$

$\displaystyle\hspace{.16 in}=\int_0^{\pi/3}(2\cos\theta-1)\,d\theta$

Answer 2

Solving $ 1- \cos \theta = \cos \theta, $ we find:

Upper limit $\pi/3 $ is OK, but the lower limit of $\theta $ should be symmetric to zero as $ -\pi/3$ . Else you will get the correct answer. May be you should revisit your sketch.