Ahhh I think I misworded that...basically I'm looking at the curves formed by the points of intersection where x > 0. They do intersect where x < 0.
Anyhow I think I did solve this issue...I simply did this by creating a line to find the area:
area = (Sqrt[3]/4) * sideLength^2;
and integrating that, while simplifying sideLength to be just the difference of the two functions.
Basically, writing another line of code to simplify things and making it so only "area" was integrated.
It seemed to work. I'm surprised though that there doesn't seem to be an easier command in Mathematica for anything like this--maybe I'm wrong?