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:
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
where is some inside the interval. This process is illustrated by figure 1.
Then, the integral would be
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
|Date of creation||2013-03-22 12:15:17|
|Last modified on||2013-03-22 12:15:17|
|Last modified by||mathwizard (128)|
|Defines||interval of integration|