Taking integrals of polar equations

Question

This is an example question that I am going through. My question is, why are they using $\frac12$ instead of $\pi$ to integrate the area?

Answer 1

Because the area of a sector is $$\color{blue}{\frac12}\color{green}r^2 \color{purple}\theta.$$

The formula comes from $\left(\frac{\theta}{2\pi}\right) \pi r^2$, hence the $\pi$ cancels out.

Hence area enclosed by $r_1(\theta) = f(\theta)$ and $r_2(\theta) = g(\theta)$, where $r_1$ is further away compared to $r_2(\theta)$, can be computed as

$$\int_{\alpha}^\beta \color{blue}{\frac12} \left[\color{green}r_1(\theta)^2-\color{green}r_2(\theta)^2\right]\, \color{purple}{d\theta} = \int_{\alpha}^\beta \color{blue}{\frac12} \left[\color{green}f(\theta)^2-\color{green}g(\theta)^2\right]\, \color{purple}{d\theta} $$