Let $f,g$ be integrable functions on $[a,b] \subset \mathbb{R} $.
If it is known that $ f(x) \leq g(x) \ \forall x \in [a,b] $, then $ \int^{a}_{b} f(x) dx \leq \int^{a}_{b} g(x) dx $.
I was wondering about the following, as it would also seem to be true, but I'm having trouble proving it. Perhaps it's not true?
If $ f(x) < g(x) \ \forall x \in [a,b] $, then is it the case that $ \int^{a}_{b} f(x) dx < \int^{a}_{b} g(x) dx $.