# definite integral

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

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