Improper integrals

Understanding Improper Integrals

  • An improper integral is a definite integral that has an infinite number of possible solutions, one or both of its limits are infinite or the integrand function is undefined at one or more points within the limits of integration.
  • They’re categorised as either Type I - where the interval of integration is infinite i.e. ∫ from a to ∞ or ∫ from -∞ to b - or Type II - where the function has an infinite discontinuity within the limits of integration.
  • Just like the general definition of an integral expresses the area under the curve, an improper integral evaluates the ‘total value’ from a function over an interval where that ‘total value’ is potentially infinite.

Evaluating Improper Integrals

  • Because we’re dealing with infinity, direct methods of evaluation can lead to indeterminate forms. To avoid this, we usually replace the infinite bound with a variable and then take a limit as that variable approaches infinity.
  • For a Type I integral such as ∫ from a to ∞ f(x) dx, evaluate it by computing the limit as t approaches ∞ of ∫ from a to t f(x) dx.
  • For a Type II integral like ∫ from a to b f(x) dx where f has a discontinuity at c, the integral is evaluated as the sum of two integrals: ∫ from a to c f(x) dx + ∫ from c to b f(x) dx where each integral is computed as a limit.
  • Occasionally, you may encounter an integral that is both Type I and Type II. In such cases, handle the infinite limits first, and then take care of the discontinuity.

Convergence and Divergence of Improper Integrals

  • Just like series, improper integrals can converge or diverge.
  • If the limit exists (is finite) when calculating an improper integral, we say the integral converges, which means it has a finite area under its curve.
  • If the limit does not exist or is infinite, we say the integral diverges, indicating an infinite area under the curve.
  • Notably, the convergence or divergence of an integral can be closely related to the convergence or divergence of a series – this is the basis of the Integral Test for series convergence.

Applications of Improper Integrals

  • Improper integrals have several applications in probability and statistics, notably in finding areas under the curve for probability density functions, especially for continuous random variables.
  • They also play a role in physics and engineering as they help in the evaluation of infinite series and solving differential equations.