integrand

(noun)

the function that is to be integrated

Related Terms

  • definite integral

Examples of integrand in the following topics:

  • Trigonometric Substitution

    • Trigonometric functions can be substituted for other expressions to change the form of integrands.
    • If the integrand contains a2+x2a^2+x^2a​2​​+x​2​​, let x=atan(θ)x = a \tan(\theta)x=atan(θ) and use the identity:
    • If the integrand contains x2−a2x^2-a^2x​2​​−a​2​​, let x=asec(θ)x = a \sec(\theta)x=asec(θ) and use the identity:
    • Example 1: Integrals where the integrand contains $a^2 − x^2$ (where aaa is positive)
    • Example 2: Integrals where the integrand contains $a^2 − x^2$ (where aaa is not zero)
  • Numerical Integration

    • The integrand f(x)f(x)f(x) may be known only at certain points, such as obtained by sampling.
    • A formula for the integrand may be known, but it may be difficult or impossible to find an antiderivative which is an elementary function.
    • Numerical integration methods can generally be described as combining evaluations of the integrand to get an approximation to the integral.
    • Also, each evaluation takes time, and the integrand may be arbitrarily complicated.
    • A 'brute force' kind of numerical integration can be done, if the integrand is reasonably well-behaved (i.e. piecewise continuous and of bounded variation), by evaluating the integrand with very small increments.
  • Trigonometric Integrals

  • Numerical Integration

    • The integrand f(x)f(x)f(x) may be known only at certain points, such as when obtained by sampling.
    • A formula for the integrand may be known, but it may be difficult or impossible to find an antiderivative which is an elementary function.
    • An example of such an integrand f(x)=exp(−x2)f(x)=\exp(-x^2)f(x)=exp(−x​2​​), the antiderivative of which (the error function, times a constant) cannot be written in elementary form.
  • Improper Integrals

    • Integrals are also improper if the integrand is undefined at an interior point of the domain of integration, or at multiple such points.
    • The problem here is that the integrand is unbounded in the domain of integration (the definition requires that both the domain of integration and the integrand be bounded).
  • Cylindrical Shells

    • In the integrand, the factor xxx represents the radius of the cylindrical shell under consideration, while  is equal to the height of the shell.
    • Therefore, the entire integrand, 2πx∣f(x)−g(x)∣dx2\pi x \left | f(x) - g(x) \right | \,dx2πx∣f(x)−g(x)∣dx, is nothing but the volume of the cylindrical shell.
  • Indefinite Integrals and the Net Change Theorem

    • f(x)f(x)f(x), the function being integrated, is known as the integrand.
  • Inverse Compton Spectra - Single Scattering

    • Depending on the value of Ef/E0E_f/E_0E​f​​/E​0​​ this integral may vanish.Specifically the integrand is non-zero only if μf\mu_fμ​f​​ lies in the range
  • Absolute Convergence and Ratio and Root Tests

    • Similarly, an improper integral of a function, ∫0∞f(x)dx\textstyle\int_0^\infty f(x)\,dx∫​0​∞​​f(x)dx, is said to converge absolutely if the integral of the absolute value of the integrand is finite—that is, if ∫0∞∣f(x)∣dx=L\int_0^\infty \left|f(x)\right|dx = L∫​0​∞​​∣f(x)∣dx=L.
  • The Physics of Bremsstrahlung

    • If ωτ≫ 1\omega \tau \gg\ 1ωτ≫ 1, the integrand will oscillate rapidly so the integral will be small.
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