Circle Area formula question Take a peek at the following proof
Everything makes sense but one thing: how did they determine that $\sqrt{\cos^2\theta}$ was positive and not negative? Thanks. 

Remember that $\sqrt{\phantom{x}}$ denotes the positive square root, so
$$\sqrt{\cos^2\theta}=|\cos\theta|\ .$$
In the paper you linked, this occurs in an integral where $\theta$ goes from $0$ to $\pi/2$.  For these $\theta$ values, $\cos\theta$ is positive, so $|\cos\theta|=\cos\theta$.
