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Half Opened Intervals

Improper integrals are defined as follows.


Suppose f is continuous on the half-open interval (a, b]. The improper integral of f from a to b is defined by the limit


If the limit exists the improper integral is said to converge. Otherwise the improper integral is said to diverge.

The improper integral can also be described in terms of definite integrals with hyperreal endpoints. We first recall that the definite integral


is a real function of two variables u and v. If u and v vary over the hyperreal numbers instead of the real numbers, the definite integral 06_applications_of_the_integral-391.giff(x) dx stands for the natural extension of D evaluated at (u, v),


Here is the description of the improper integral using definite integrals with hyperreal endpoints.

Let f be continuous on (a, b].

(1) 06_applications_of_the_integral-393.gif= S if and only if 06_applications_of_the_integral-394.gif ≈ S for all positive infinitesimal ε.

(2) 06_applications_of_the_integral-395.gif= ∞ (or -∞) if and only if 06_applications_of_the_integral-396.gif is positive infinite (or negative infinite) for all positive infinitesimal s.

Example 1
Example 2
Example 3

Notice that we use the same symbol for both the definite and the improper integral. The theorem below justifies this practice.

Last Update: 2006-11-05