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. 

In the proof that you cite there is an integral from $0$ to $\frac{\pi}{2}$, which for a circle means that it is in in the first quadrant (which makes sense, considering they were looking for the area of the quarter circle formed in the first quadrant), and because the  $y$ values are positive, we use a positive square root. 
For a more graphic representation take a look at this: 

In essence, the circle is actually two functions when put on a plane like this, and we are interested in the top funtion (i.e. $y \geq 0$). We are integrating the first quadrant, which means that our integral if of the form $$\int\limits_0^{\frac{\pi}{2}} f(\theta)d\theta$$
now, because we know that that results in an integral with a square root, we chose the positive root because we are integrating the upper function, and therefore our $f(\theta) \geq 0$.
