Let $F ( x ) = \int _ { x } ^ { x ^ { 2 } + \frac { \pi } { 6 } } 2 \cos ^ { 2 } t \, d t$ for all $x \in \mathbb { R }$ 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 $F ^ { \prime } ( a ) + 2$ is the area of the region bounded by $x = 0 , y = 0 , y = f ( x )$ and $x = a$, then $f ( 0 )$ is