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Calculus III

Course: Calculus I
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 ...

 The First Fundamental Theorem of Calculus (FTC1) If a function f is continuous on [a,b] and F is an antiderivative of f on the interval [a,b], then int_a^b f(x) dx = F(b) - F(a).

 The Second Fundamental Theorem of Calculus (FTC2) If f is continuous on an open interval I containing a, then for every x in the interval, d/dx [int_a^x f(x) dx] = f(x).

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

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)

rev. 2020-11-27