Improper Integral

Definition

An improper integral is a type of integral that has a domain that extends to infinity or contains a point where the integrand is not defined. These integrals are used to study the convergence or divergence of infinite series and sequences.

5 Must Know Facts For Your Next Test

1. Improper integrals can be classified as Type I (when the interval of integration extends to infinity) or Type II (when the integrand is not defined at a point within the interval of integration).
2. The Divergence Test states that if the improper integral $\int_{a}^{\infty} f(x) dx$ diverges, then the series $\sum_{n=a}^{\infty} f(n)$ also diverges.
3. The Integral Test states that if the improper integral $\int_{a}^{\infty} f(x) dx$ converges, then the series $\sum_{n=a}^{\infty} f(n)$ also converges.
4. Improper integrals can be evaluated using techniques such as integration by parts, substitution, or the comparison test.
5. The convergence or divergence of an improper integral is determined by the behavior of the integrand as the variable approaches the point of discontinuity or the limits of integration.

Review Questions

• Explain the relationship between improper integrals and the convergence or divergence of infinite series.
  • Improper integrals are closely related to the convergence or divergence of infinite series. The Divergence Test states that if an improper integral diverges, then the corresponding infinite series also diverges. Conversely, the Integral Test states that if an improper integral converges, then the corresponding infinite series also converges. This connection allows mathematicians to study the behavior of infinite series by analyzing the properties of related improper integrals, which can be more tractable to evaluate.
• Describe the two main types of improper integrals and how they are evaluated.
  • Improper integrals can be classified into two main types: Type I and Type II. Type I improper integrals have a domain that extends to infinity, such as $\int_{a}^{\infty} f(x) dx$. These integrals are evaluated by examining the behavior of the integrand as the variable approaches the limit of integration. Type II improper integrals contain a point where the integrand is not defined, such as $\int_{a}^{b} f(x) dx$ where $f(x)$ is not defined at $x = c$ for $a < c < b$. These integrals are evaluated by considering the limits as the variable approaches the point of discontinuity from both sides.
• Analyze how the Divergence Test and Integral Test can be used to determine the convergence or divergence of infinite series related to improper integrals.
  • The Divergence Test and Integral Test provide a powerful framework for studying the convergence or divergence of infinite series by relating them to the properties of improper integrals. The Divergence Test states that if an improper integral diverges, then the corresponding infinite series also diverges. Conversely, the Integral Test states that if an improper integral converges, then the corresponding infinite series also converges. This allows mathematicians to leverage the techniques used to evaluate improper integrals, such as integration by parts, substitution, or the comparison test, to determine the behavior of infinite series. By establishing these connections, the convergence or divergence of a wide range of mathematical objects can be systematically analyzed.

Related terms

Convergence:

The property of a series or sequence where the partial sums approach a finite limit as the number of terms increases.

Divergence:

The property of a series or sequence where the partial sums do not approach a finite limit as the number of terms increases.

Integral Test:

A method used to determine the convergence or divergence of a series by comparing it to the integral of a related function.