Let $$F(x) = \int_{x}^{x^2 + \frac{\pi}{6}} 2 \cos^2t dt$$ for all $$x \in R$$ and $$f : \left[0, \frac{1}{2}\right] \rightarrow [0, \infty)$$ be a continuous function. For $$a \in \left[0, \frac{1}{2}\right]$$, if $$F' (a) + 2 $$ is the area of the region bounded by x = 0, y = 0, y = f(x) and x = a, then f(0) is