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:

\int_{a}^{b}f(x)\ dx.

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

For example, use a Riemann sum 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 $R_{n}$ be this approximation, which can be written as

R_{n}=\sum_{i=1}^{n}f(x_{i}^{*})\Delta x,

where $x_{i}^{*}$ is some $x$ inside the $i^{\rm th}$ interval. This process is illustrated by figure 1.

Figure 1: The area under the graph approximated by rectangles

Then, the integral would be

\int_{a}^{b}f(x)\ dx=\lim_{n\to\infty}R_{n}=\lim_{n\to\infty}\sum_{i=1}^{n}f(x% _{i}^{*})\Delta x.

This limit does not necessarily exist for every function $f$ and it may depend on the particular choice of the $x_{i}^{*}$ . If all those limits coincide and are finite, then the integral exists. This is true in particular for continuous $f$ .

Furthermore we define

\int_{b}^{a}f(x)\ dx=-\int_{a}^{b}f(x)\ dx.

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

$\displaystyle\int_{a}^{b}(f(x)+g(x))\ dx$	$\displaystyle=$	$\displaystyle\int_{a}^{b}f(x)\ dx+\int_{a}^{b}g(x)\ dx;$
$\displaystyle\int_{a}^{b}(f(x)-g(x))\ dx$	$\displaystyle=$	$\displaystyle\int_{a}^{b}f(x)\ dx-\int_{a}^{b}g(x)\ dx;$
$\displaystyle\int_{a}^{b}f(x)\ dx$	$\displaystyle=$	$\displaystyle\int_{a}^{c}f(x)\ dx+\int_{c}^{b}f(x)\ dx;$
$\displaystyle\int_{a}^{b}cf(x)\ dx$	$\displaystyle=$	$\displaystyle c\int_{a}^{b}f(x)\ dx.$

There are other generalizations about integrals, but many require the fundamental theorem of calculus.

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