integrand

An integrand is the function that is being integrated in an integral. In the standard single-variable setting, a definite integral is written as ∫_a^b f(x) dx, where f(x) is the integrand. For an indefinite integral, ∫ f(x) dx, the aim is to find an antiderivative of f. The integrand may depend on several variables, as in f(x, y) or f(x, y, z).

The integral sign encodes the process of accumulating infinitesimal contributions of the integrand over its domain. The value of a definite integral is a number that depends on the limits of integration; an indefinite integral yields a family of antiderivatives differing by a constant.

Common examples include the integrand x^2 in ∫_0^1 x^2 dx = 1/3. The integrand can be more complex, such as a function with exponential, logarithmic, or trigonometric factors. The possibility of a finite integral depends on the integrability of the function; Riemann integrability applies to certain bounded functions on a closed interval, while Lebesgue integrability extends to a broader class of functions.

Multivariable problems use integrands like f(x, y) integrated over a region R with dA, or f(x, y, z) integrated with dV. Improper integrals arise when the integrand is unbounded or the domain is unbounded; convergence is defined as the limit of a family of proper integrals.

Applications are widespread, including probability theory, physics, and engineering. The integrand determines what is being accumulated, while the integral computes the total value, such as area, probability, or expected quantities.