Chapter 6

Chapter 6 Overview

The need to calculate instantaneous rates of change led the discoverers of calculus to an investigation of the slopes of tangent lines and, ultimately, to the derivative—to what we call differential calculus. But derivatives revealed only half the story. In addition to a calculation method (a “calculus”) to describe how functions change at any given instant, they needed a method to describe how those instantaneous changes could accumulate over an interval to produce the function that describes the total change.

Early in the 14th century, a group of scholars at Merton College in Oxford, England, began exploring how to find the accumulated distance from knowledge of the velocity. In 1638, Galileo Galilei used their ideas to explain the motion of falling bodies and justify his claim that the earth circles the sun. Later that century, Isaac Newton studied these accumulation functions and, in 1666, discovered a remarkable insight that connected accumulation functions to what he already knew about derivatives. If there was one moment when calculus was born, this was it.

Today, what we call the integral calculus or integration has two distinct interpretations. We begin this chapter by looking at integration as accumulation. But it also can be viewed as reversing the process of differentiation, what we call antidifferentiation. Newton’s insight, that these two are connected, is what is called the Fundamental Theorem of Calculus (also known as the Fundamental Theorem of Integral Calculus).