definite integral

The definite integral with respect to x of some function f(x) over the compact interval [a,b] with a<b, the interval of integration, is defined to be the “area under the graph of f(x) with respect to x” (if f(x) is negative, then you have a negative area). The numbers a and b are called lower and upper limit respectively. It is written as:


One way to find the value of the integral is to take a limit of an approximation technique as the precision increases to infinityMathworldPlanetmathPlanetmath.

For example, use a Riemann sumMathworldPlanetmath which approximates the area by dividing it into n intervals of equal widths, and then calculating the area of rectangles with the width of the interval and height dependent on the function’s value in the interval. Let Rn be this approximation, which can be written as


where xi* is some x inside the ith interval. This process is illustrated by figure 1.

Figure 1: The area under the graph approximated by rectangles

Then, the integral would be


This limit does not necessarily exist for every function f and it may depend on the particular choice of the xi*. If all those limits coincide and are finite, then the integral exists. This is true in particular for continuousMathworldPlanetmathPlanetmath f.

Furthermore we define


We can use this definition to arrive at some important properties of definite integrals (a, b, c are constant with respect to x):

ab(f(x)+g(x))𝑑x = abf(x)𝑑x+abg(x)𝑑x;
ab(f(x)-g(x))𝑑x = abf(x)𝑑x-abg(x)𝑑x;
abf(x)𝑑x = acf(x)𝑑x+cbf(x)𝑑x;
abcf(x)𝑑x = cabf(x)𝑑x.

There are other generalizationsPlanetmathPlanetmath about integrals, but many require the fundamental theorem of calculusMathworldPlanetmathPlanetmath.

Title definite integral
Canonical name DefiniteIntegral
Date of creation 2013-03-22 12:15:17
Last modified on 2013-03-22 12:15:17
Owner mathwizard (128)
Last modified by mathwizard (128)
Numerical id 16
Author mathwizard (128)
Entry type Definition
Classification msc 26A06
Related topic AreaOfPlaneRegion
Related topic IntegralsOfEvenAndOddFunctions
Related topic IntegralOverAPeriodInterval
Defines interval of integration
Defines upper limit
Defines lower limit