Integral mean value theorem

Let $f:[a,b]\to \mathbb{R}$ be Riemann integrable. Let

Then there exists a number $\mu \in [m,M]$ such that

If, in addition, $f$ has the Darboux property (intermediate value property; for example if $f$ is continuous or if $f$ is a derivative), then there exists $\xi \in [a,b]$ such that

Related

• Riemann integral
• Fundamental theorem of calculus