Summary of the Mean Value Theorem for Integrals

If $f(t)$ is integrable on the interval $[a,b]$, with $a < b$, put:

$$A(f) = \frac{1}{b - a}\int_a^b f(t) dt.$$

• $A(f)$ is called the average or mean value of $f$ on the interval $[a,b]$.
• If $f$ is continuous, then $A(f) = f(\xi)$ for some $\xi$ in $(a, b)$.
• If $f(t) \ge 0$ then $A(f) \ge 0$.
• If $f(t) \ge 0$ and $f$ is continuous, then $A(f) > 0$, unless $f(t) = 0$, for all $t$ in $[a,b]$.