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Calculus III

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

Overview

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.

Objectives

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

• 4.5.1 Use graphical symmetry to evaluate a definite integral
• 4.5.2 Use the knowledge about a function being odd or even (and the consequential symmetries) to evaluate a definite integral
• 4.5.3 Understand the hypothesis and conclusion of the mean value theorem for integrals and be able to apply it
• 4.5.4 Understand the hypothesis and conclusion of the average value theorem and be able to apply it

Terminology

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

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