Calculus Essentials For Dummies

Chapter 8

Introduction to Integration

IN THIS CHAPTER

Integrating — adding it all up

Approximating areas

Using the definite integral to get exact areas

If you’re still reading, I presume you survived differentiation (Chapters 4–7). Now you begin the second major calculus topic: integration. Just as two simple ideas lie at the heart of differentiation — rate (like miles per hour) and the slope of a curve — integration can also be understood in terms of two simple ideas: adding up small pieces of something and the area under a curve.

Integration: Just Fancy Addition

Say you want to determine the volume of the lamp’s base in Figure 8-1. Because no formula for the volume of such a weird shape exists, you can’t calculate the volume directly. You can, however, calculate the volume with integration. Imagine that the base is cut up into thin, horizontal slices, as shown on the right in Figure 8-1.

FIGURE 8-1: A lamp with a curvy base and the base cut into thin horizontal slices.

Do you see how each slice is shaped like a thin pancake? Now, because there is a formula for the volume of a pancake ...