Unit 11 - Sections 5.3 to 5.5

Module Overview

With this unit, we learn about the Fundamental Theorem of Calculus. The great result of the field, it lets us see that integrals and derivatives are (as far as possible) inverse operations. This is a grand result and deserves our careful attention. We also cover the method of substitution, which is the most-used of the ways in which we do integrals (other than, for some people, Wolfram Alpha).

Learning Outcomes

Section 5.3: It is called the Fundamental Theorem of Calculus for a reason - in this section, we put together everything we have done for the last few chapters and find that the answer to the question of the area under the graph is precisely the answer to the question of the inverse of the derivative. This is the most important result in the entire subject and simply knowing it is nowhere near as useful as understanding it. Take your time with this section and its proofs.
Section 5.4: A couple of small results are collected in this section. The average value of a function will be the most useful one for our purposes and, luckily, it is quite a simple idea.
Section 5.5: The reverse of the chain rule, substitution is an absolutely necessary rule in integration. It is difficult to get through the next chapter and impossible to get through Chapter 7 without it, so make sure to do a lot of exercises in order to become comfortable with this material.

Type: Video Links
Title	Author	Description
The Fundamental Theorem of Calculus \| The Fundamental Theorem of Calculus	Shai Cohen
Using the Fundamental Theorem of Calculus \| Using the Fundamental Theorem of Calculus	Shai Cohen
Integrals of Even and Odd Functions \| Integrals of Even and Odd Functions	Shai Cohen
Finding Integrals Example \| Finding Integrals Example	Shai Cohen
Derivatives of Integrals Example \| Derivatives of Integrals Example	Shai Cohen
Average Value of a Function \| Average Value of a Function	Shai Cohen
The Mean Value Theorem for Integrals \| The Mean Value Theorem for Integrals	Shai Cohen
Average Value Example \| Average Value Example	Shai Cohen
Substitution - Indefinite Integrals \| Substitution - Indefinite Integrals	Shai Cohen
Substitution - Definite Integrals \| Substitution - Definite Integrals	Shai Cohen
Substitution Example - Complicated \| Substitution Example - Complicated	Shai Cohen
Substitution Example - Definite and Indefinite \| Substitution Example - Definite and Indefinite	Shai Cohen
Substitution Example - Different Methods \| Substitution Example - Different Methods	Shai Cohen