Improper Integrals: Discontinuous Integrands

A definite integral becomes improper when its integrand is discontinuous at one of the limits of integration. This occurs when the function is undefined or becomes infinite at an endpoint, creating a vertical asymptote at the boundary. The region under the curve becomes unbounded, requiring a limiting process to evaluate the integral properly.

Replace the discontinuous endpoint with a variable, then evaluate the definite integral over the interval where the function is defined. Take the limit as the variable approaches the original endpoint. If this limit exists and is finite, the improper integral converges, indicating the total area remains bounded despite the singular behavior.

Vertical asymptotes at integration boundaries cause the integrand to become infinite at those points, making the integral improper. The function's unbounded behavior near the asymptote creates a discontinuity that must be handled through a limiting process. This approach allows the contribution of the region near the discontinuity to be examined carefully.

The limiting process replaces the problematic limit with a variable and evaluates the integral over an interval where the function is defined. As the variable approaches the original endpoint, the limit captures the full behavior of the function near the discontinuity. This method determines whether a finite accumulated quantity can still be obtained despite the singular behavior.

Consider a radial electric field that becomes infinite at the center and decreases with distance. To compute the electric potential difference from near the center to distance R, replace the lower limit with a small positive value and evaluate the integral. Taking the limit as this value approaches zero yields a finite electric potential difference despite the discontinuity at the origin.

Convergence occurs when the limit of the improper integral exists and is finite. This indicates that despite the function being undefined or infinite at an endpoint, the total area under the curve remains bounded. If the limit does not exist or is infinite, the improper integral diverges, meaning no finite accumulated quantity can be obtained.

Improper integrals with discontinuous integrands have singularities at finite endpoints where the function becomes infinite or undefined. In contrast, improper integrals infinite intervals extend over unbounded domains. Both require limiting processes to evaluate, but they address different types of unbounded behavior in integration problems.