iint

iint is a notation used to denote an iterated or double integral in ASCII or informal mathematical text. It represents the idea of performing two single-variable integrations in sequence, which together form the two-dimensional integral of a function over a region. The standard mathematical form is the double integral ∬ f(x,y) dA, but iint emphasizes the order of integration and the two-step process.

In practice, iint is read as performing an inner integral with respect to one variable, followed by an outer integral with respect to the other. The limits for each integration are specified for the corresponding variable. For a rectangle like 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1, an iint expression might be written as iint_0^1 iint_0^1 f(x,y) dy dx, which corresponds to ∫_0^1 ∫_0^1 f(x,y) dy dx. The order can be swapped if the conditions of Fubini’s theorem are satisfied, yielding ∫_0^1 ∫_0^1 f(x,y) dx dy.

Common uses include computing area, mass, probability, or any quantity that requires integrating a function over a two-dimensional region. The iint notation is often found in ASCII math, textbooks, or computer algebra systems that prefer plain-text representations, but it is less formal than the explicit ∬ symbol in polished mathematics. In published work, the double integral symbol ∬ is preferred, while iint serves as a convenient shorthand in non-typeset materials.

Related concepts include single integrals (int or ∫) and multiple integrals of higher order, which extend the idea to three or more dimensions.