Let $S$ be the set of all continuous functions $f: [0, 1] \to \mathbb{R}$. Consider two subsets of $S$: $A = \{ f \in S : f(x) \ge 0 \text{ for all } x \in [0, 1] \}$ $B = \{ f \in S : \int_0^1 f(x) dx = 0 \}$ Which of the following statements about $A \cap B$ is true?