integrand

(noun)

the function that is to be integrated

Related Terms

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 $a^2+x^2$, let $x = a \tan(\theta)$ and use the identity:
    • If the integrand contains $x^2-a^2$, let $x = a \sec(\theta)$ and use the identity:
    • Example 1: Integrals where the integrand contains $a^2 − x^2$ (where $a$ is positive)
    • Example 2: Integrals where the integrand contains $a^2 − x^2$ (where $a$ is not zero)

  • Numerical Integration

    • The integrand $f(x)$ 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 $f(x)$ 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 $f(x)=\exp(-x^2)$, 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 $x$ represents the radius of the cylindrical shell under consideration, while  is equal to the height of the shell.
    • Therefore, the entire integrand, $2\pi x \left | f(x) - g(x) \right | \,dx$, is nothing but the volume of the cylindrical shell.

  • Indefinite Integrals and the Net Change Theorem

    • $f(x)$, the function being integrated, is known as the integrand.

  • 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 $\int_0^\infty \left|f(x)\right|dx = L$.