[Math] Area of the region inside $r = 1 – \cos(\theta)$ and also inside $r = \cos(\theta)$

Question

Pretty simple polar integration question that I've been having trouble with…

The question says it all. I identified the limits of integration by setting $1 – \cos(\theta) = \cos(\theta)$ so that $\cos(\theta) = \frac{1}{2}$ and $\theta = \pm \frac{\pi}{3}$.

I've tried the integral

$$ \frac{1}{2}\int_{-\frac{\pi}{3}}^{\frac{\pi}{3}} \cos^2(\theta) – (1 – \cos(\theta))^2 = \sqrt{3} – \frac{\pi}{3}$$

Back of the book says the answer is $\frac{7\pi}{12} – \sqrt{3}$.

I think the polar curves are messing me up. Am I interpreting $r = \cos(\theta)$ wrong or something? Do I need to change the bounds of integration?

Answer 1

You are calculating the area in the following graph

and your calculation is correct.

But the question asks for the area "inside" both graphs which can be seen below:

This area is then $$A=\int_0^{\pi/3}(1-\cos x)^2dx+\int_{\pi/3}^{\pi/2}\cos^2xdx$$ which is what the back of your book says.