# Chapter Five

# ACCUMULATED CHANGE: THE DEFINITE INTEGRAL

Contents

5.1 Distance and Accumulated Change

A Thought Experiment: How Far Did the Car Go?

Visualizing Distance on the Velocity Graph

Visualizing Distance on the Velocity Graph: Area Under Curve

Approximating Total Change from Rate of Change

Improving the Approximation: The Role of *n* and Δ*t*

Writing Left- and Right-Hand Sums Using Sigma Notation

Taking the Limit to Obtain the Definite Integral

Estimating a Definite Integral from a Table or Graph

Rough Estimates of a Definite Integral

5.3 The Definite Integral as Area

Relationship Between Definite Integral and Area

5.4 Interpretations of the Definite Integral

The Notation and Units for the Definite Integral

5.5 Total Change and the Fundamental Theorem of Calculus

Marginal Cost & Change in Total Cost

The Definite Integral as an Average

How to Visualize the Average on a Graph

PROJECTS: Carbon Dioxide in Pond Water, Flooding in the Grand Canyon

Theorems About Definite Integrals

## 5.1 DISTANCE AND ACCUMULATED CHANGE

In Chapter 2, we used the derivative to find the rate of change of a function. Here ...

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