In mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, f(x, y) or f(x, y, z). Integrals of a function of two variables over a region in (the real-number plane) are called double integrals, and integrals of a function of three variables over a region in (real-number 3D space) are called triple integrals. For multiple integrals of a single-variable function, see the Cauchy formula for repeated integration.
In mathematical analysis, an improper integral is an extension of the notion of a definite integral to cases that violate the usual assumptions for that kind of integral. In the context of Riemann integrals (or, equivalently, Darboux integrals), this typically involves unboundedness, either of the set over which the integral is taken or of the integrand (the function being integrated), or both. It may also involve bounded but not closed sets or bounded but not continuous functions.
In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of Göttingen in 1854, but not published in a journal until 1868. For many functions and practical applications, the Riemann integral can be evaluated by the fundamental theorem of calculus or approximated by numerical integration, or simulated using Monte Carlo Integration.
In mathematics, the Riemann–Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes. The definition of this integral was first published in 1894 by Stieltjes. It serves as an instructive and useful precursor of the Lebesgue integral, and an invaluable tool in unifying equivalent forms of statistical theorems that apply to discrete and continuous probability.
In mathematics, the Peano–Jordan measure (also known as the Jordan content) is an extension of the notion of size (length, area, volume) to shapes more complicated than, for example, a triangle, disk, or parallelepiped. It turns out that for a set to have Jordan measure it should be well-behaved in a certain restrictive sense. For this reason, it is now more common to work with the Lebesgue measure, which is an extension of the Jordan measure to a larger class of sets.
