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Theorem 1: Arbitrary Infinitesimals


Given a continuous function f on [a, b] and two positive infinitesimals dx and du, the definite integrals with respect to dx and du are the same,


From now on when we write a definite integral 04_integration-89.gif, it is understood

that dx is a positive infinitesimal. By Theorem 1, it doesn't matter which infinitesimal.

The proof of Theorem 1 is based on the following intuitive idea. Figure 4.2.1 shows the two Riemann sums04_integration-90.gif and 04_integration-91.gif. We see from the figure that the difference04_integration-92.gif is a sum of rectangles of infinitesimal height. These difference rectangles all lie between the horizontal lines y = -ε and y = ε, where ε is the largest height. Thus


Taking standard parts,



Figure 4.2.1

Theorem 1 shows that whenever Δx is positive infinitesimal, the Riemann sum is infinitely close to the definite integral,


This fact can also be expressed in terms of limits. It shows that the Riemann sum approaches the definite integral as Δx approaches 0 from above, in symbols


Given a continuous function f on an interval I, Theorem 1 shows that the definite integral is a real function of two variables a and b,


We now formally define the area as the definite integral shown in Figure 4.2.2.


Figure 4.2.2


If f is continuous and f (x) ≥ 0 on [a, b], the area of the region below the curve y = f(x) from a to b is defined as the definite integral:


The next two theorems give basic properties of the integral.

Last Update: 2010-11-26