There are two uses of the term ``integral'' in calculus. One is as a method of finding areas, extending the ``method of exhaustion'' described in the Introduction. The other is as a synonym of the antiderivative of a function. What should be surprising is that these two ideas are intertwined, in that you find areas by using antiderivatives. However, that connection is commonly the only way in which areas are actually computed, so the power of the theorem that gives the connection between them is often overlooked. But that theorem, the Fundamental Theorem of Calculus, or FTC, is the foundation of the development of calculus (hence the name, of course). The discovery of this result, independently and approximately at the same time by both Issac Newton and G. Leibniz, is the reason those two mathematicians are given credit as the inventors of calculus. The basic ideas of the derivative were known before Newton and Leibniz, and the method of exhaustion goes back to classical Greek mathematicians, but the connection between the two was the key to developing the analytical theory and applications that we now call calculus.
The integral as ``area under a curve'' (the definite integral) is defined, and computed in the few cases where it can reasonably be found from the definition. Properties of the definite integral are described
Here we state and prove the FTC, which is the theoretical high point of the course.
Since the FTC implies that the way to compute definite integrals involves antiderivatives, which are usually called indefinite integrals, here we find the first methods to systematically compute them.
It is in general much more difficult to compute (indefinite) integrals than derivatives. In this section we derive and apply the three most basic techniques of integration, substitution, integration by parts, and partial fractions. there are other techniques which we will not discuss, postponing them to the next semester.
Copyright (c) 2000 by David L. Johnson.