User:Cronholm144/Integral
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In calculus, the integral of a function extends the concept of an ordinary sum. While an ordinary sum is taken over a discrete set of values, integration extends this concept to sums over continuous domains. The process of evaluating (or determining) an integral is known as integration. Integration is used to find the "total amount" of a property, whenever that property varies or is distributed in a known manner across a continuous domain. For example, instantaneous velocity changes moment to moment through the continuous domain of time. To sum up all the instantaneous velocities over a given interval of time, and hence obtain the total displacement that occurred, we evaluate the integral of the velocity over the given interval of time. Though this concept was the starting point for the development of integration theory by Newton and Leibniz, it has since been extended and newer definitions stress different aspects.
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If a function has an integral, it is said to be integrable. The function for which the integral is calculated is called the integrand. The region over which a function is being integrated is called the domain of integration. In general the integrand may be a function of more than one variable, and the domain of integration may be an area, volume, a higher dimensional region, or even an abstract space that does not have a geometric structure in any usual sense. The simplest case, the integral of a real-valued function f of one real variable x on the interval [a, b], is denoted:
The ∫ sign represents integration; a and b are the lower limit and upper limit of integration, defining the domain of integration; f(x) is the integrand; and dx is a notation for the variable of integration.
This form of integral may be identified with the signed area under the curve defined by the graph of f over the interval [a,b].
The term "integral" may also refer to antiderivatives. Though they are closely related through the fundamental theorem of calculus, the two notions are conceptually distinct. When one wants to clarify this distinction, an antiderivative is referred to as an indefinite integral (a function), while the integrals discussed in this article are termed definite integrals.