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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 ...