PlanetMath: definite integral

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definite integral

(Definition)

The definite integral with respect to of some function over the compact interval with , the interval of integration, is defined to be the “area under the graph of with respect to ” (if is negative, then you have a negative area). The numbers and are called lower and upper limit respectively. It is written as:

$\displaystyle \int_a^bf(x) \ dx .$

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 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 be this approximation, which can be written as

$\displaystyle R_n = \sum_{i=1}^{n} f(x_i^*) \Delta x ,$

where

is some

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

$\displaystyle \int_a^bf(x) \ dx = \lim_{n \to \infty} R_n = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*) \Delta x .$

This limit does not necessarily exist for every function

and it may depend on the particular choice of the

. If all those limits coincide and are finite, then the integral exists. This is true in particular for continuous

Furthermore we define

$\displaystyle \int_b^af(x)\ dx=-\int_a^bf(x)\ dx.$

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

$\displaystyle \int_a^b(f(x) + g(x)) \ dx$	$\displaystyle =$	$\displaystyle \int_a^bf(x)\ dx + \int_a^bg(x)\ dx;$
$\displaystyle \int_a^b(f(x) - g(x)) \ dx$	$\displaystyle =$	$\displaystyle \int_a^bf(x)\ dx - \int_a^bg(x)\ dx ;$
$\displaystyle \int_a^bf(x) \ dx$	$\displaystyle =$	$\displaystyle \int_a^cf(x)\ dx + \int_c^bf(x)\ dx ;$
$\displaystyle \int_a^bcf(x) \ dx$	$\displaystyle =$	$\displaystyle c\int_a^bf(x)\ dx.$