MVT for Integrals

• Let $a,b\in R$
• Let $a<b$
• If $f$ is cont on $[a,b]$ Then $\exists c\in (a,b)$ s.t $f(c)=\frac{1}{b-a}{\int }_a^{b}f(x)dx$
• Or $\exists c\in (a,b)$ s.t $\frac{F(b)-F(a)}{b-a}=f(c)$
• Kathleen likes this form: $\exists c\in [a,b]$ s.t ${\int }_a^{b}f(x)dx=f(c)(b-a)$

Intuition

Proof