Improper Integral

Definition

An improper integral is a type of integral that occurs when either the limits of integration are infinite or the integrand becomes unbounded within the interval of integration. This concept is crucial for evaluating integrals that cannot be computed using standard techniques, as it allows mathematicians to analyze areas under curves that extend indefinitely or have discontinuities.

5 Must Know Facts For Your Next Test

1. Improper integrals can be classified into two main types: those with infinite limits of integration and those with integrands that have vertical asymptotes within the interval.
2. To evaluate an improper integral, you often replace it with a limit, transforming it into a proper integral which can then be solved.
3. The convergence or divergence of an improper integral depends on the behavior of the integrand near the points where it becomes unbounded or at infinity.
4. Common examples of improper integrals include $\,\int_{1}^{\infty} \frac{1}{x^2} dx$ and $\,\int_{0}^{1} \frac{1}{x} dx$, which require careful limit evaluation.
5. The comparison test is a useful technique for determining the convergence of an improper integral by comparing it with another known convergent or divergent integral.

Review Questions

• How do you determine whether an improper integral converges or diverges?
  • To determine if an improper integral converges or diverges, you first need to evaluate the integral by rewriting it as a limit. For example, if you're evaluating $\,\int_{a}^{\infty} f(x) dx$, you would express it as $\,\lim_{b \to \infty} \int_{a}^{b} f(x) dx$. After computing this limit, if it approaches a finite number, the integral converges; if it approaches infinity or does not exist, then the integral diverges.
• Explain how to evaluate an improper integral with a vertical asymptote at one of its bounds.
  • To evaluate an improper integral that has a vertical asymptote at one of its bounds, such as $\,\int_{a}^{b} f(x) dx$ where $f(x)$ approaches infinity at $x = c$ (with $a < c < b$), you would split the integral into two parts: $\,\int_{a}^{c} f(x) dx$ and $\,\int_{c}^{b} f(x) dx$. Each part must be evaluated as a limit. For example, for the first part, use $\,\lim_{t \to c^-} \int_{a}^{t} f(x) dx$ and similarly for the second part. If both limits exist and are finite, you can add them together for the total value of the improper integral.
• Critically analyze the role of comparison tests in evaluating improper integrals and their implications in mathematical analysis.
  • Comparison tests play a crucial role in evaluating improper integrals by providing a method to assess their convergence without directly calculating them. The idea is to compare an improper integral to another known integral whose convergence is already established. If $0 \leq f(x) \leq g(x)$ for all $x$ in the interval and if $\int g(x) dx$ converges, then $\int f(x) dx$ must also converge. Conversely, if $g(x)$ diverges, so does $f(x)$. This technique not only simplifies evaluations but also helps in understanding the behavior of functions within different bounds, significantly impacting mathematical analysis and applications.

Related terms

Convergence: Convergence refers to the behavior of an integral or series that approaches a finite limit as the input or terms increase without bound.

Divergence: Divergence describes the situation where an integral or series fails to converge, leading to an infinite or undefined result.

Limit: A limit is a fundamental concept in calculus that describes the value that a function approaches as the input approaches a particular point or infinity.