improper integral

Examples of improper integral in the following topics:

• Improper Integrals

  • Specifically, an improper integral is a limit of one of two forms.
  • First, an improper integral could be a limit of the form:
  • Second, an improper integral could be a limit of the form:
  • However, the improper integral does exist if understood as the limit
  • Evaluate improper integrals with infinite limits of integration and infinite discontinuity

• The Integral Test and Estimates of Sums

  • The integral test is a method of testing infinite series of nonnegative terms for convergence by comparing them to an improper integral.
  • The integral test for convergence is a method used to test infinite series of non-negative terms for convergence.
  • The infinite series $\sum_{n=N}^\infty f(n)$ converges to a real number if and only if the improper integral $\int_N^\infty f(x)\,dx$ is finite.
  • In other words, if the integral diverges, then the series diverges as well.
  • The integral test applied to the harmonic series.

• Comparison Tests

  • The limit comparison test is a method of testing for the convergence of an infinite series, while the direct comparison test is a way of deducing the convergence or divergence of an infinite series or an improper integral by comparison with other series or integral whose convergence properties are already known.
  • The direct comparison test provides a way of deducing the convergence or divergence of an infinite series or an improper integral.
  • In both cases, the test works by comparing the given series or integral to one whose convergence properties are known.

• Absolute Convergence and Ratio and Root Tests

  • Similarly, an improper integral of a function, $\textstyle\int_0^\infty f(x)\,dx$, is said to converge absolutely if the integral of the absolute value of the integrand is finite—that is, if $\int_0^\infty \left|f(x)\right|dx = L$.

• Iterated Integrals

  • An iterated integral is the result of applying integrals to a function of more than one variable.
  • An iterated integral is the result of applying integrals to a function of more than one variable (for example $f(x,y)$ or $f(x,y,z)$) in such a way that each of the integrals considers some of the variables as given constants.
  • If this is done, the result is the iterated integral:
  • Similarly for the second integral, we would introduce a "constant" function of $x$, because we have integrated with respect to $y$.
  • Use iterated integrals to integrate a function with more than one variable

• Line Integrals

  • A line integral is an integral where the function to be integrated is evaluated along a curve.
  • A line integral (sometimes called a path integral, contour integral, or curve integral) is an integral where the function to be integrated is evaluated along a curve.
  • The function to be integrated may be a scalar field or a vector field.
  • This weighting distinguishes the line integral from simpler integrals defined on intervals.
  • The line integral finds the work done on an object moving through an electric or gravitational field, for example.

• Double Integrals Over Rectangles

  • The multiple integral is a type of definite integral extended to functions of more than one real variable—for example, $f(x, y)$ or $f(x, y, z)$.
  • Formulating the double integral , we first evaluate the inner integral with respect to $x$:
  • We could have computed the double integral starting from the integration over $y$.
  • Double integral as volume under a surface $z = x^2 − y^2$.
  • The rectangular region at the bottom of the body is the domain of integration, while the surface is the graph of the two-variable function to be integrated.

• Change of Variables

  • One makes a change of variables to rewrite the integral in a more "comfortable" region, which can be described in simpler formulae.
  • The limits of integration are often not easily interchangeable (without normality or with complex formulae to integrate).
  • One makes a change of variables to rewrite the integral in a more "comfortable" region, which can be described in simpler formulae.
  • When changing integration variables, however, make sure that the integral domain also changes accordingly.
  • Use a change a variables to rewrite an integral in a more familiar region

• Integration Using Tables and Computers

  • Tables of known integrals or computer programs are commonly used for integration.
  • Integration is the basic operation in integral calculus.
  • We also may have to resort to computers to perform an integral.
  • A compilation of a list of integrals and techniques of integral calculus was published by the German mathematician Meyer Hirsch as early as in 1810.
  • Computers may be used for integration in two primary ways.

• Numerical Integration

  • Numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite integral.
  • This article focuses on calculation of definite integrals.
  • The basic problem considered by numerical integration is to compute an approximate solution to a definite integral:
  • If $f(x)$ is a smooth well-behaved function, integrated over a small number of dimensions and the limits of integration are bounded, there are many methods of approximating the integral with arbitrary precision.
  • There are several reasons for carrying out numerical integration.