I am working on the following problem:
Let $f$ and $g$ be continuous on $[a,b]$ such that $\int_a^b f(x) \ dx = \int_a^b g(x) \ dx.$ Prove that there is a $c \in [a,b]$ so that $f(c) = g(c)$.
Not too sure how to proceed with this. I tried moving everything into one integral as follows:
$$ \int_a^b f(x)-g(x) \ dx = 0. $$
Not quite sure where to go with this. Perhaps I need to apply the Intermediate Value Theorem? Rolles Theorem?