integral

—integrality, n. —integrally, adv.

/in"ti greuhl, in teg"reuhl/, adj.

1. of, pertaining to, or belonging as a part of the whole; constituent or component: integral parts.

2. necessary to the completeness of the whole: This point is integral to his plan.

3. consisting or composed of parts that together constitute a whole.

4. entire; complete; whole: the integral works of a writer.

5. Arith. pertaining to or being an integer; not fractional.

6. Math. pertaining to or involving integrals.

n.

7. an integral whole.

8. Math.

a. Also called Riemann integral. the numerical measure of the area bounded above by the graph of a given function, below by the x-axis, and on the sides by ordinates drawn at the endpoints of a specified interval; the limit, as the norm of partitions of the given interval approaches zero, of the sum of the products of the function evaluated at a point in each subinterval times the length of the subinterval.

b. a primitive.

c. any of several analogous quantities. Cf. improper integral, line integral, multiple integral, surface integral.

[1545-55; < ML integralis. See INTEGER, -AL¹]

Syn. 2. essential, indispensable, requisite.

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I

Fundamental concept of calculus related to areas and other quantities modeled by functions.

A definite integral gives the area between the graph of a function and the horizontal axis between vertical lines at the endpoints of an interval. It also calculates the net change in a system over an interval, thus leading to formulas for the work done by a varying force or the distance traveled by an object moving at varying speeds. When only the function is given, with no interval, it is known as an indefinite integral. The process of solving either a definite or an indefinite integral is called integration. According to the fundamental theorem of calculus, a definite integral can be calculated by using its antiderivative (a function whose rate of change, or derivative, equals the function being integrated). Integrals extend to higher dimensions through multiple integrals. See also line integral; surface integral.

II

(as used in expressions)

integral calculus

integral equation

integral transform

line integral

multiple integral

surface integral

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▪ calculus

      in mathematics, either a numerical value equal to the area under the graph of a function for some interval (definite integral) or a new function the derivative of which is the original function (indefinite integral). These two meanings are related by the fact that a definite integral of any function that can be integrated can be found using the indefinite integral and a corollary to the fundamental theorem of calculus. The definite integral (also called Riemann integral) of a function f(x) is denoted as

      (see integration [for symbol]) and is equal to the area of the region bounded by the curve (if the function is positive between x = a and x = b) y = f(x), the x-axis, and the lines x = a and x = b. An indefinite integral, sometimes called an antiderivative, of a function f(x), denoted by

      is a function the derivative of which is f(x). Because the derivative of a constant is zero, the indefinite integral is not unique. The process of finding an indefinite integral is called integration.

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