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Course: Calculus I
Topic: Antiderivatives
Subtopic: Theorems of Calculus (FTC, MVT for Integrals, AV)

Overview

This lesson includes theorems that form the guts of integral calculus. There are two Fundamental Theorems of Calculus (FTC) both connecting differentiation to integration. The Mean Value Theorem (MVT) for Integrals and its consequence the Average Value (AV) Theorem provide a major purpose of integrals. Be sure you understand the theorems and how to apply them.

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 basic properties and rules of definite integrals
• 4.4.4 Understand the hypothesis and conclusion of the first fundamental theorem of calculus, the second fundamental theorem of calculus, the mean value theorem for integrals, and the average value theorem
• 4.4.5 Apply the first fundamental theorem of calculus, the second fundamental theorem of calculus, the mean value theorem for integrals, and the average value theorem

Terminology

Define: integrable, integrability, First Fundamental Theorem of Calculus (FTC1), Second Fundamental Theorem of Calculus (FTC2), Mean Value Theorem for Integrals, Average Value Theorem

Supplemental Resources (recommended)

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

Supplemental Resources (optional)