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, isdefined 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 techniqueas the precision increases to infinity.

For example, use a Riemann sum which approximatesthe area by dividing it into $n$ intervals of equal widths, and then calculating the areaof 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.