fundamental theorem of calculus for Riemann integration
In this entry we discuss the fundamental theorems of calculus for Riemann integration.
- Let be a continuous function in an interval and a Riemann integrable function such that except at most in a finite number of points . Then .
Notice that the second fundamental theorem is not a converse of the first. In the first we conclude that except in a set of measure zero (http://planetmath.org/MeasureZeroInMathbbRn), while in the second we assume that except in a finite number of points. In fact, the two theorems can never be the converse of each other as the following example shows:
Example : Let be the devil staircase function, defined on . We have that
is continuous in ,
is clearly a Riemann integrable function and .
This leads to the question: what kind functions can be expressed as , for some function ? The answer to this question lies in the concept of absolute continuity (http://planetmath.org/AbsolutelyContinuousFunction2) (a which the devil staircase does not possess), but for that a more general of integration must be developed (the Lebesgue integration (http://planetmath.org/Integral2)).
|Title||fundamental theorem of calculus for Riemann integration|
|Date of creation||2013-03-22 17:57:32|
|Last modified on||2013-03-22 17:57:32|
|Last modified by||asteroid (17536)|
|Defines||first fundamental theorem of calculus (Riemann integral)|
|Defines||second fundamental theorem of calculus (Riemann integral)|