Course: Calculus I
Topic: Antiderivatives
Subtopic: Mean Value Theorem for Integrals and Average Value


This lesson includes two theorems that are consequences of the Fundamental Theorem of Calculus. The Mean Value Theorem (MVT) for Integrals and its consequence the Average Value (AV) Theorem provide further purpose to integrals from a visual perspective.


By the end of this topic you should know and be prepared to be tested on:


Terms you should be able to define: symmetry, odd/even function (and graphical significence)

The Mean Value Theorem for Integrals

If `f` is continuous on `[a,b]`, then there exists a number `c` in `[a,b]` 
such that `int_a^b f(x) dx = (b-a) * f(c)`. In other words,
`f" continuous on "[a,b] => EE c in [a,b]` `int_a^b f(x) dx = (b-a) * f(c)` 

The Average Value Theorem of a Function on an Interval

If `f` is integrable on `[a,b]`, then the average value
of `f` on the interval is `f_(ave) = 1/(b-a) int_a^b f(x) dx` .

Mini-Lectures and Examples

STUDY: Calculus Theorems (MVT, AV)

Supplemental Resources (recommended)

Useful video from 3Blue1Brown's The Essence of Calculus: What does area have to do with slope?

Supplemental Resources (optional)

Application of Average Value: Head Injury Criterion (HIC): Severity Index continued at Head Injury Criterion (HIC): HIC Index, example

rev. 2020-11-28