Integral Range of Bounded Derivative Function

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function such that $f(0)=0$, $f(1)=1$, and $\left|f^{\prime}(x)\right| \leq 2$ for all real numbers $x$. If $a$ and $b$ are real numbers such that the set of possible values of $\int_{0}^{1} f(x) d x$ is the open interval $(a, b)$, determine $b-a$.