Definite Integral

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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 Definite Integral. 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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If a definite integral is desired, one must use the fact that
Other books on integrals involving Bessel functions mostly discuss definite integrals, and though he also treats definite integrals, he emphasizes indefinite ones.
While it is a good practice after obtaining a definite integral to find its derivative as a check that the manipulations were carried out correctly, and a nice reminder of the relationship between the functions concerned, this can sometimes be tedious and at other times can be extremely tedious, so is frequently not done by time-poor or symbolic manipulation-averse students.
As the screen suggests, this particular device represents an indefinite integral by leaving blanks where the limits of a definite integral might appear.
Then the field is open to deal with anti-derivatives and solve simple differential equations, well before the definite integral is introduced through sequences of Riemann sums.