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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. DEFINITION Let f be a continuous function on an interval I and let a < b be two points in I. Let dx be a positive infinitesimal. Then the definite integral of f from a to b with respect to dx is defined to be the standard part of the infinite Riemann sum with respect to dx, in symbols We also define 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, x^{2}t 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.


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