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In calculus, the integral of a function is a generalization of area, mass, volume, sum, and total. Unlike the process of differentiation, there are several different definitions of integration, all of which have different technical underpinnings. However, any two different ways of integrating a function will give the same result if they are both defined.

Intuitively, the integral of a continuous, positive real-valued function f of one real variable x between a left endpoint a and a right endpoint b represents the area bounded by the lines x=a, x=b, the x-axis, and the curve defined by the graph of f. More formally, if we let S={(x,y):a≤x≤b,0≤y≤f(x)}, then the integral of f between a and b is the measure of S.

As an example, if f is the constant function f(x)=3, then the integral of f between 0 and 10 is the area of the rectangle bounded by the lines x=0, x=10, y=0, and y=3. The area is 10c, so the value of the integral is 30.

Integrals can be taken over regions other than intervals. In general, the integral over a set E of a function f is written ∫_Ef(x)dx. Here x need not be a real number, but, for instance, a vectorA vector in physics and engineering typically refers to a quantity that has close relationship to the spatial coordinates, informally described as an object with a "magnitude" and a "direction". The word vector is also now used for more general concepts ( in R³. Fubini's theoremStatement In mathematical analysis, Fubini's theorem named in honor of Guido Fubini, states that if : the integral being taken with respect to a product measure on the space over , then : the first two integrals being iterated integrals, and the third bei shows that such integrals can be rewritten as an iterated integral. In other words, the integral can be calculated by integrating one coordinate at a time.

If a function has an integral, it is said to be integrable. The function for which the integral is calculated is called the integrand. Integrals are sometimes called definite integrals to emphasize that they result in a number, not another function. This is to distinguish them from indefinite integrals, which are another name for an antiderivativeIn calculus, an antiderivative or primitive function of a given real valued function f is a function F whose derivative is equal to f i. F ' f''. The process of finding antiderivatives is antidifferentiation (or indefinite integration . For example: F ''x. If the domain of the function is the real numberIn mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. The term "real number" is a retronym coined in response to " imaginary number". Real numbers mays, and if the region of integration is an intervalTopology In elementary algebra, an interval is a set that contains every real number between two indicated numbers, and possibly the two numbers themselves. Interval notation is where the permitted values for a variable are expressed as ranging over an in, then the greatest lower boundIn mathematics the infimum of a subset of some set is the greatest element that is smaller than all other elements of the subset. Consequently the term greatest lower bound is also commonly used. Infima of real numbers are a common special case that is es of the interval is called the lower limit of integration, and the least upper boundIn mathematics, the supremum of a given set is the least element which is greater than or equal to each element of the set. Consequently, it is also referred to as the least upper bound . In general, unless a set contains a greatest element, the supremum is called the upper limit of integration.

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