Improper Integral

Definition

An improper integral is a type of integral where the domain of integration extends to infinity or the integrand becomes unbounded at one or more points within the interval of integration. These integrals are used to model physical phenomena that involve infinite or unbounded quantities.

5 Must Know Facts For Your Next Test

1. Improper integrals are classified into two types: integrals with infinite limits of integration and integrals with discontinuous integrands.
2. To evaluate an improper integral, one must first determine whether the integral converges or diverges, and then use appropriate techniques to calculate the value of the integral.
3. Convergence of an improper integral is determined by the behavior of the integrand near the points of discontinuity or at the infinite limits of integration.
4. The comparison test and the limit comparison test are commonly used to establish the convergence or divergence of improper integrals.
5. Improper integrals are important in various fields, such as physics, engineering, and mathematical analysis, where they are used to model physical phenomena with infinite or unbounded quantities.

Review Questions

• Explain the difference between a Riemann integral and an improper integral, and provide an example of each.
  • A Riemann integral is defined over a finite interval, where the integrand is bounded and the domain of integration is closed and bounded. For example, the integral $\int_a^b f(x) dx$, where $a$ and $b$ are finite real numbers, and $f(x)$ is a continuous function on the interval $[a, b]$, is a Riemann integral.\n\nIn contrast, an improper integral is a type of integral where the domain of integration extends to infinity or the integrand becomes unbounded at one or more points within the interval of integration. For example, the integral $\int_0^\infty \frac{1}{x} dx$ is an improper integral, as the integrand becomes unbounded at $x = 0$.
• Describe the two main types of improper integrals and explain how the convergence or divergence of each type is determined.
  • The two main types of improper integrals are:\n\n1. Integrals with infinite limits of integration: These are integrals of the form $\int_a^\infty f(x) dx$ or $\int_{-\infty}^b f(x) dx$, where $a$ and $b$ are finite real numbers. The convergence or divergence of these integrals is determined by the behavior of the integrand $f(x)$ as $x$ approaches infinity.\n\n2. Integrals with discontinuous integrands: These are integrals of the form $\int_a^b f(x) dx$, where $f(x)$ is discontinuous at one or more points within the interval $[a, b]$. The convergence or divergence of these integrals is determined by the behavior of the integrand $f(x)$ at the points of discontinuity.
• Explain how the comparison test and the limit comparison test can be used to establish the convergence or divergence of improper integrals, and provide an example of each.
  • The comparison test and the limit comparison test are commonly used to determine the convergence or divergence of improper integrals.\n\nThe comparison test states that if $\int_a^\infty f(x) dx$ converges and $0 \leq g(x) \leq f(x)$ for all $x \geq a$, then $\int_a^\infty g(x) dx$ also converges. Conversely, if $\int_a^\infty g(x) dx$ diverges and $g(x) \leq f(x)$ for all $x \geq a$, then $\int_a^\infty f(x) dx$ also diverges.\n\nThe limit comparison test states that if $\lim_{x\to\infty} \frac{f(x)}{g(x)} = L$, where $L$ is a finite, non-zero constant, then $\int_a^\infty f(x) dx$ converges if and only if $\int_a^\infty g(x) dx$ converges. For example, to determine the convergence of $\int_1^\infty \frac{1}{x^p} dx$, we can use the limit comparison test with $g(x) = \frac{1}{x}$, and find that the integral converges if $p > 1$ and diverges if $p \leq 1$.

Related terms

Convergent Integral: An improper integral that has a finite value, meaning the integral converges.

Divergent Integral: An improper integral that does not have a finite value, meaning the integral diverges.

Riemann Integral: A standard integral defined over a finite interval, where the integrand is bounded and the domain of integration is closed and bounded.