I'm about to give a first-semester calculus lecture covering the mean value theorem for integrals:
If $f$ is continuous on $[a,b]$, then there is some $c\in(a,b)$ such that $(b-a)f(c)=\int_a^b f(x)\,dx$.
In past semesters, I've shown examples in which I confirm that this theorem holds for some specific $f(x)$ and $[a,b]$, by solving for $c\in(a,b)$. But this is just checking the theorem -- not actually applying it. A "real" application occurs, for example, in the proof of Taylor's remainder formula, but my students aren't ready for that example.
What is a good "real" application of this theorem, suitable for students in first-semester calculus?