# definite integral

The *definite integral* with respect to $x$ of some function $f(x)$ over the compact interval $[a,b]$ with $$, 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:

$${\int}_{a}^{b}f(x)\mathit{d}x.$$ |

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

$${R}_{n}=\sum _{i=1}^{n}f({x}_{i}^{*})\mathrm{\Delta}x,$$ |

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

Then, the integral would be

$${\int}_{a}^{b}f(x)\mathit{d}x=\underset{n\to \mathrm{\infty}}{lim}{R}_{n}=\underset{n\to \mathrm{\infty}}{lim}\sum _{i=1}^{n}f({x}_{i}^{*})\mathrm{\Delta}x.$$ |

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

$${\int}_{b}^{a}f(x)\mathit{d}x=-{\int}_{a}^{b}f(x)\mathit{d}x.$$ |

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

${\int}_{a}^{b}}(f(x)+g(x))\mathit{d}x$ | $=$ | ${\int}_{a}^{b}}f(x)\mathit{d}x+{\displaystyle {\int}_{a}^{b}}g(x)\mathit{d}x;$ | ||

${\int}_{a}^{b}}(f(x)-g(x))\mathit{d}x$ | $=$ | ${\int}_{a}^{b}}f(x)\mathit{d}x-{\displaystyle {\int}_{a}^{b}}g(x)\mathit{d}x;$ | ||

${\int}_{a}^{b}}f(x)\mathit{d}x$ | $=$ | ${\int}_{a}^{c}}f(x)\mathit{d}x+{\displaystyle {\int}_{c}^{b}}f(x)\mathit{d}x;$ | ||

${\int}_{a}^{b}}cf(x)\mathit{d}x$ | $=$ | $c{\displaystyle {\int}_{a}^{b}}f(x)\mathit{d}x.$ |

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 |