Chapter 4: The Fundamental Theorem of Calculus

Chapters 2 and 3 focused on three core concepts of calculus:

  • Limits

  • Area under a curve
    (using Riemann sums)

  • Slope of a curve
    (with slope functions and derivatives)

In this chapter, you will recognize and explore the fundamental connections among these three core concepts. Together, these concepts will define the Fundamental Theorem of Calculus, a powerful tool that will be useful for calculating the exact area under a curve.

Chapter Goals

Set up and evaluate integrals to
calculate areas.

Create general area functions.

Investigate properties of definite
integrals.

Discover and use the Fundamental
Theorem of Calculus.

Calculate area between curves.

Chapter Outline

Section 4.1

You will be introduced to the concept of integrals as the limit of Riemann sums. You will learn to evaluate these limits on your calculator and geometrically (when possible). You will also develop a list of the properties of integrals.

Section 4.2

You will use algebra and geometry to discover area functions. You will create area functions to calculate the area under a curve between a fixed number and a variable endpoint. You will recognize the relationship between derivatives and integrals. You will apply the Fundamental Theorem of Calculus.

Section 4.3

You will work with data in a mock trial allowing you to apply the skills learned in the first two sections of this chapter.

Section 4.4

You will be using multiple strategies to calculate the area between curves, deciding which strategy is best for each situation.

Section 4.5

You will learn Newton’s Method to approximate roots.