Introduction

There are two main ways to think of the definite integral. The easiest one to understand, and the one we'll consider first, is as a means for computing areas (and volumes). The second way the definite integral is used is as a sum. That is, we use the definite integral to ``add things up''. Here are some examples.

• Finding the mass of a body in two or three dimensions, whose density is not a constant.
• Computing net or total distance traveled by a moving object.
• Computing work involved in moving an object, compressing a gas, or pumping a liquid.
• Computing average values, e.g. centroids and centers of mass, moments of inertia, and averages of probability distributions.

Of course, when we use a definite integral to compute an area or a volume, we are adding up area or volume. So you might ask why make a distinction? The answer is that the notion of an integral as a means of computing an area or volume is much more concrete and is easier to understand.

We will learn in class that the definite integral is actually defined as a (complicated) limit of sums, so it makes sense that the integral should be thought of as a sum. We will also learn in class that the indefinite integral, or anti-derivative, can be used to evaluate definite integrals. Students often concentrate on techniques for evaluating integrals, and ingnore the definition of the integral as a sum. This is a mistake, for the following reasons.

1. Many functions don't have anti-deriviatives that can be written down as formulas. Definite integrals of such function must be done using numerical techniques, which always depend on the definition of the integral as a sum.
2. In many applications of the integral in engineering and science, you aren't given the function which is to be integrated and must derive it. The derivation always depends on the definition of the integral as a sum.