Oneigenlijke integralen: oneindige intervallen

An integral is considered improper because of an infinite interval when the upper or lower limit of integration extends to infinity, creating an unbounded region under the curve.

In this case, the infinite bound is replaced with a variable, and the integral is evaluated by taking the limit as that variable approaches infinity.

This method helps find whether the total area under the curve stays finite even across an infinite domain.

A practical example is calculating the total integrated intensity of light that passes through a medium, such as fog, over an infinite distance, assuming the medium is uniform.

In such cases, light intensity decreases with distance, following an exponential decay pattern.

To find the total integrated intensity, the infinite upper limit is first replaced with a variable, t.

The intensity function is then integrated from zero to t, and the limits of integration are substituted to give an expression that includes an exponential term.

The final step is taking the limit as t approaches infinity, where the exponential term approaches zero, leaving a finite value.

This confirms that the total integrated intensity — the area under the curve — can stay finite even with an infinite domain.