Mean Value Theorem for Integrals

If f is continuous on the closed interval [a, b],then there exists a number c in the closed interval [a, b] such that $\int_a^b f(x)dx=f(c)(b-a)$

• Proof
• Reference

Average Value of a Function

Definition of the Average Value of a Function on an Interval: If $f$ is integrable on the closed interval [a, b], then the average value of $f$ on the interval is

$$ f(c)=\frac 1 {b-a}\int_a^bf(x)dx $$

By the equation of Mean Value Theorem for Integrals.

• Average value is the y value when multiple by $\Delta x$ is equal to the integral from a to b of$f(x) dx$