Calculus Through Areas.

Let $f : R \rightarrow R$ be a differentiable function such that $f(0) = 0, f(1) = 1$ and $\left| f’\left( x \right) \right| \le 2$ $\forall x \in R$ . Further if $a$ and $b$ are real numbers such that the set of possible values of

$\displaystyle \large\ \int _{ 0 }^{ 1 }{ f\left( x \right) } dx$

is the open interval $(a, b)$ , then find $(b - a)$ .