The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Definite Integral

one of the fundamental concepts of mathematical analysis; the solution of a number of problems in geometry, mechanics, and physics reduces to a definite integral. The definite integral is a number equal to the limit of the sums of a particular type (integral sums) corresponding to a function f(x) and an interval [a, b]; it is denoted by . Geometrically, the definite integral expresses the area of a “curvilinear trapezoid” bounded by the interval [a, b] on the x-axis, the graph of the function f(x), and the ordinates of the points on the graph that have abscissas a and b. For a precise definition and generalization of the definite integral, seeINTEGRAL and INTEGRAL CALCULUS.

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equals p.sub.4 / [the definite integral from negative infinity to infinity of] (f.sub.B of ([the integer portion of] x + 0.5) dx / f.sub.B of(M)

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Thus it is equal to the definite integral [[integral].sup.b.sub.0] 1 + 1 + x dx, which can be calculated by elementary calculus (see [3] and [4]):

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As can be seen, the numerical integration methods are able to approximate a value for a definite integral using the values of the function at points within the interval of the integrand.

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Generally, the primitive of a Lagrange 1-form L(t,x(t),[??](t))dt can be written either as (definite integral)

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Multipy [T.sub.m](x)/[square root of([1 - [chi square]] (m = 0, 1, ...) to the two sides of (1) and definite integral from--1 to 1.

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