Let $f(x) = 3ax^{2} + (1 - a)$ be a real coefficient polynomial function, where $-\frac{1}{2} \leq a \leq 1$. On the coordinate plane, let $\Gamma$ be the region enclosed by $y = f(x)$ and the $x$-axis for $-1 \leq x \leq 1$.
Prove that for all $a \in \left[-\frac{1}{2}, 1\right]$, the area of $\Gamma$ is always 2. (Non-multiple choice question, 2 points)