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See also Integration (general).
In mathematics the term integral has two unrelated meanings, one relating to integers, and the other relating to the integral calculus.
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Integral value
A real number is called integral if it is an integer. The integral value of a real number x is defined to be the largest integer which is less than or equal to x; it is often denoted by ⌊x⌋ and also called the floor function.
Integral of a mathematical function
In the integral calculus, the integral of a function is informally defined as the size of the area delimited by the x axis and the graph of the function. In the case of non-negative functions, the notion of area is the usual one. For functions which take negative values, a special interpretation is used, and "negative area" is possible.
Let f(x) be a function of the interval [a,b] into the real numbers. For simplicity, assume that this function is non-negative (it takes no negative values.) The set S=Sf:={(x,y)|0≤y≤f(x)} is the region of the plane between f and the x axis. Measuring the "area" of S is desirable, and this area is denoted by ∫f, and it is the (definite) integral of f.
Details can be found under Riemann integral and Lebesgue integral. The concept of Riemann integration was developed first, and Lebesgue integrals were developed to deal with pathological cases for which the Riemann integral was not defined. If a function is Riemann integrable, then it is also Lebesgue integrable, and the two integrals coincide.
The antiderivative approach occurs when we seek to find a function F(x) whose derivative F(x) is some given function f(x). This approach is motivated by calculus, and is the main method used for calculating the area under the curve as described in the preceding paragraph, for functions given by formulae.
Functions which have antiderivatives are also Riemann integrable (and hence Lebesgue integrable.) The nonobvious theorem that states that the two approaches ("area under the curve" and "antiderivative") are in some sense the same as the fundamental theorem of calculus
The nuance between Riemann and Lebesgue integration
Both the Riemann and the Lebesgue integral are approaches to integration which seek to measure the area under the curve, and the overall schema in both cases is the same.
First, we select a family of elementary functions, for which we have an obvious way of measuring the area under the curve. In the case of the Riemann integral, this choice is so that the area under the curve can be regarded as a finite union of rectangles, and the functions are called step functions. For the Lebesgue integral, "rectangle" is replaced by something more sophisticated, and the resulting functions are called simple functions.
Then we try to impose monotonicity. If 0≤f≤g (and hence Sf is a subset of Sg) then we should have that ∫f≤∫g. With this monotonicity requirement, for an arbitrary non-negative function f, we can approximate its area from below using a carefully chosen elementary function s (in the case of Riemann integration, a step function, and in the case of Lebesgue integration, a simple function.) We choose s so that s≤f but s is very close to f. The area under s is a lower bound for the integral of f, and it is called a lower sum. In the case of the Riemann integral, we also produce upper sums in a similar fashion: we choose step functions, say s, so that s≥f but s is very close to f, and we regard such an upper sum as an upper bound for the area under f. The Lebesgue theory does not use upper sums.
Lastly, a limit-taking step is taken to make the elementary functions approach f more and more closely, and an area is obtained for some functions f. The functions which we can integrate are said to be integrable. However, the differences begin here; the Riemann theory was simpler thus far, but its simplicity results in a more limited set of integrable functions than the Lebesgue theory. In addition, the interaction between limits and the integral are more difficult to describe in the Riemann setting.
Other integrals
Although the Riemann and Lebesgue integrals are the most important ones, a number of others exist, including but not limited to:
- the Darboux integral[?], a variation of the Riemann integral
- the Denjoy integral[?], an extension of both the Riemann and Lebesgue integrals
- the Euler integral
- the Haar integral[?]
- the Henstock-Kurzweil integral[?], an extension of both the Riemann and Lebesgue integrals (also called HK-integral)
- the Henstock-Kurzweil-Stieltjes integral[?] (also called HK-Stieltjes integral)
- the Lebesgue-Stieltjes integral[?] (also called Lebesgue-Radon integral)
- the Perron integral[?], which is equivalent to the restricted Denjoy integral[?]
- the Stieltjes integral[?], an extension of the Riemann integral (also called Riemann-Stieltjes integral)
See also: Calculus, List of integrals.