integral test

Examples of integral test in the following topics:

• 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.
  • In other words, if the integral diverges, then the series diverges as well.
  • Although we won't go into the details, the proof of the test also gives the lower and upper bounds:
  • The integral test applied to the harmonic series.

• Tips for Testing Series

  • Convergence tests are methods of testing for the convergence or divergence of an infinite series.
  • Convergence tests are methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence, or divergence of an infinite series.
  • When testing the convergence of a series, you should remember that there is no single convergence test which works for all series.
  • But if the integral diverges, then the series does so as well.
  • The integral test applied to the harmonic series.

• Convergence of Series with Positive Terms

  • For these general cases, we can experiment with several well-known convergence tests (such as ratio test, integral test, etc.).
  • We will learn some of these tests in the following atoms.

• Comparison Tests

  • Comparison test may mean either limit comparison test or direct comparison test, both of which can be used to test convergence of a series.
  • Comparison tests may mean either limit comparison tests or direct 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 $\int_0^\infty \left|f(x)\right|dx = L$.
  • The ratio test states that,
  • The root test is a criterion for the convergence (a convergence test) of an infinite series.
  • The root test was developed first by Augustin-Louis Cauchy and so is sometimes known as the Cauchy root test, or Cauchy's radical test.
  • The root test states that

• Improper Integrals

  • An Improper integral is the limit of a definite integral as an endpoint of the integral interval approaches either a real number or $\infty$ or $-\infty$.
  • Such an integral is often written symbolically just like a standard definite integral, perhaps with infinity as a limit of integration.
  • It is often necessary to use improper integrals in order to compute a value for integrals which may not exist in the conventional sense (as a Riemann integral, for instance) because of a singularity in the function, or an infinite endpoint of the domain of integration.
  • However, the Riemann integral can often be extended by continuity, by defining the improper integral instead as a limit:
  • Evaluate improper integrals with infinite limits of integration and infinite discontinuity

• 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