Let $f$ be a differentiable function from $\mathbb{R}$ to $\mathbb{R}$ such that $|f(x) - f(y)| \le 2|x - y|^{3/2}$, for all $x, y \in \mathbb{R}$. If $f(0) = 1$, then $\int_0^1 f^2(x) \, dx$ is equal to: