## You are here

Homedefinite integral

## Primary tabs

# 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.

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.

## Mathematics Subject Classification

26A06*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections

## Info

## Attached Articles

## Corrections

math mode by drini ✓

parentheses by pahio ✓

terms by pahio ✓

lower and upper limit by pahio ✓

## Comments

## Additional Reference

PlanetMath article: Non-Newtonian calculus.