Improper Integrals: Infinite Intervals

An integral is classified as improper due to an infinite interval when at least one of its limits of integration extends to positive or negative infinity. In such cases, the region under the curve is unbounded, and standard techniques for evaluating definite integrals are not directly applicable. Instead, the improper integral is defined through a limiting process that allows one to determine whether the accumulated area remains finite despite the infinite domain.

Application to Exponential Decay Models

A practical application of improper integrals over infinite intervals arises in modeling exponential decay processes. One example is the total integrated intensity of light passing through a uniform medium, such as fog over an infinite distance. In this scenario, the light intensity decreases exponentially with distance and can be modeled by a function of the form I₀​e^−kx, where I₀​ is the initial intensity and k > 0 is a decay constant.

To compute the total integrated intensity, the integral

is first rewritten as

Evaluating the integral yields an expression containing an exponential term e^−kt. As t approaches infinity, this term approaches zero, leaving a finite result. This demonstrates that even though the distance is infinite, the total integrated intensity remains finite due to the exponential decay of the function.