Improper Integrals: Discontinuous Integrands

Evaluating Areas Under Curves with Discontinuities

A definite integral is considered improper when the integrand is discontinuous at one of the limits of integration. This occurs when the function is undefined or becomes infinite at an endpoint, making the corresponding region under the curve unbounded. Such behavior is commonly associated with vertical asymptotes at the boundary of the interval. To properly define and evaluate these integrals, a limiting process is used to determine whether a finite accumulated quantity can still be obtained.When a function is discontinuous at an endpoint, the integral is redefined by replacing the problematic limit with a variable. The resulting definite integral is evaluated over an interval where the function is defined, and a limit is then taken as the variable approaches the original endpoint. This method allows the contribution of the region near the discontinuity to be examined carefully. If the limit exists and is finite, the improper integral converges, indicating that the total area under the curve remains bounded despite the singular behavior.

As an example, consider a hypothetical radial electric field that depends on distance from the origin and becomes infinite at the center. The field decreases with distance but creates a discontinuity at the lower limit of integration. To compute the electric potential difference from near the center to a distance R, the lower limit is replaced with a small positive value. The integral is evaluated over this interval, and the limit is taken as the lower bound approaches zero. This process accounts for the singular behavior at the origin and yields a finite electric potential difference despite the discontinuity.