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.
A definite integral becomes improper when its integrand is discontinuous at an endpoint. This means the function is undefined or infinite at that point.
This discontinuity usually happens when the function has a vertical asymptote at the boundary.
In such cases, the area under the curve is found by replacing the discontinuous endpoint with a variable and evaluating the integral using a limit.
This concept can be used to calculate the electric potential difference for a hypothetical radial electric field, starting from a point near the center and going out to a distance R.
The electric field is modeled by a function proportional to one over square root of r. This function decreases with distance but becomes infinite near the center.
This creates an unbounded region near the origin, with a vertical asymptote at the lower limit.
To calculate the potential difference from the center outward, the discontinuous lower limit is replaced with a small positive value t. The integral is then set up from t to R and evaluated.
Taking the limit as t approaches zero captures the full behavior of the electric field.
This approach calculates the electric potential difference while resolving the discontinuity at the origin.
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