Definite Integral

We are now ready to define the central concept of this chapter, the definite integral. Recall that the derivative was defined as the standard part of the quotient Δy/Δx and was written dy/dx. The "definite integral" will be defined as the standard part of the infinite Riemann sum

and is written . Thus the Δx is changed to dx in analogy with our differential

notation. The Σ is changed to the long thin S, i.e., , to remind us that the integral is obtained from an infinite sum. We now state the definition carefully.

By this definition, for each positive infinitesimal dx the definite integral

is a real function of two variables defined for all pairs (u, w) of elements of I. The symbol x is a dummy variable since the value of

does not depend on x.

In the notation for the Riemann sum and for the

integral, we always use matching symbols for the infinitesimal dx and the dummy variable x. Thus when there are two or more variables we can tell which one is the dummy variable in an integral. For example, x²t can be integrated from 0 to 1 with respect to either x or t. With respect to x,

(where dx = 1/H), and we shall see later that

With respect to t, however,

and we shall see later that

The next two examples evaluate the simplest definite integrals. These examples do it the hard way. A much better method will be developed in Section 4.2.

Integral of a constant function and the area of a rectangle.

Integral of a straight line.