Topic: Antiderivatives

Subtopic: Fundamental Theorems of Calculus

**Overview**

The __Fundamental Theorem of Calculus (FTC)__ is key to making the connections between differentiation and integration. There are actually two forms of FTC. Undertanding FTC requires algebraic skills and visual observations. Supporting content is also included here, eg. properties of integrals.

**Objectives**

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

- 4.4.1 Know that continuity implies integrability and that the converse is not true
- 4.4.2 Recognize a function as integrable or not
- 4.4.3 Apply properties and rules of integrals
- 4.4.4 Be able to compute net area including from a given graph or table of data
- 4.4.5 Recognize the integral as an area function
- 4.4.6 Understand the hypothesis and conclusion of the first and second fundamental theorem of calculus
- 4.4.7 Apply the first and second fundamental theorem of calculus

**Terminology**

Terms you should be able to define: integrable, integrability, net area, and ...

If a function f is continuous on [a,b] and F is an antiderivative |

If f is continuous on an open interval I containing a, |

Note: I'm using the traditional titles of FTC1 and FTC2, but some authors reverse them or call them by different names. Several authors/websites call FTC2 "Fundamental Theorem of Calculus (part 1)" and FTC1 "Fundamental Theorem of Calculus (part 2)" including the Briggs/Cochran text.

**Mini-Lectures and Examples**

STUDY: Properties of Integrals, Net Area, Area Function, and FTC

**Supplemental Resources (recommended)**

Explore area as a function with The Area Function applet.

Two more applets for exploration: FTC 1 (theoretical part) and FTC 2 (practical part)

Useful video from 3Blue1Brown's *The Essence of Calculus*: Integration and the FTC

**Supplemental Resources (optional)**

Video: The Integral, Selwyn Hollis's Video Calculus

Video: The Fundamental Theorem of Calculus, Selwyn Hollis's Video Calculus

Lesson: Properties of the Definite Integral, Dale Hoffman's Contemporary Calculus

Lesson: The Fundamental Theorem of Calculus, Dale Hoffman's Contemporary Calculus

rev. 2020-11-27