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Elementary Calculus Integral The Definite Integral Theorem: Infinite Riemann Sum
 

Theorem: Infinite Riemann Sum

THEOREM 1

Let f be a continuous function on an interval I, let a < b be two points in I, and let dx be a positive infinitesimal. Then the infinite Riemann sum

04_integration-43.gif

is a finite hyperreal number.

PROOF

Let B be a real number greater than the maximum value of f on [a,b].

04_integration-45.gif

Figure 4.1.12

Consider first a real number Dx > 0. We can see from Figure 4.1.12 that the finite Riemann sum is less than the rectangular area B · (b - a);

04_integration-49.gif

Therefore by the Transfer Principle,

04_integration-46.gif

In a similar way we let C be less than the minimum of f on [a, b] and show that

04_integration-50.gif

Thus the Riemann sum 04_integration-51.gif is finite.

Last Update: 2006-Nov-06