integrando

Integrando is the term used in calculus to denote the function that is to be integrated. In standard notation, if an integral is written as ∫ f(x) dx, the function f(x) is the integrand (integrando in some languages). The integral then represents the accumulation of the values of this function over a specified domain, yielding either an antiderivative in the indefinite case or a numeric value in the definite case.

Notation and variable of integration: The integrand is the expression inside the integral sign. When the integral is ∫ f(x) dx, f(x) is the integrand and x is the variable of integration. If the integral has limits, ∫_a^b f(x) dx, the integrand remains f(x). The result should not depend on the particular letter used for the variable, since it is a dummy variable.

Multiple variables and domains: The integrand can depend on several variables, and integration may occur over a region or with respect to a chosen variable, for example ∫_D f(x,y) dx dy, where the integration is taken over the domain D. In such contexts, the integrand represents the density or rate to be accumulated over the region.

Examples: In ∫_0^2 x^2 dx, the integrand is x^2. In ∫_0^1 e^(−x^2) dx, the integrand is e^(−x^2). The integrand is distinct from the integral itself; the former is a function, the latter the operator yielding a value or function.

Broader context: The concept extends beyond the Riemann integral to generalized frameworks such as Lebesgue integrals and various path or surface integrals used in physics and engineering.