Riemann integral

The Riemann integral ( Riemann integral) is a named after the German mathematician Bernhard Riemann method for clarifying the philosophical idea of the area of ​​between the axis and the graph of a function. The Riemann integral term includes not only the more general Lebesgue to the two classical calculus. In many applications, only integrals of continuous or piecewise continuous functions are required. Then the somewhat simpler, but less general term of the integral control functions is sufficient.

The concept of the Riemann integral underlying is to approach this area with the help of easy to calculate surface area of rectangles. It is followed in that there are two families of rectangles so chooses in each step that the graph of the function "between" is them. By successively increasing the number of rectangles is obtained with time an increasingly accurate approximation of the graph by belonging to the rectangles step functions. Accordingly, the area between the graph and the axis through the areas of the rectangles can be approximated.

• 2.1 Lebesgue criterion
• 2.2 Examples

Definitions

There are two common methods for defining the Riemann integral:

• Jean Gaston Darboux attributed to the method by means of upper and lower sums and
• Riemann's original method by means of Riemann sums.

The two definitions are equivalent: Each function is integrable in the sense of Darboux when she is integrable in the Riemann sense; In this case, match the values ​​of the two integrals. In typical Analysis introductions, especially in school, now largely the Darbouxsche formulation is used to define. Riemann sums often occur added as a further aid, such as to prove the fundamental theorem of integral and differential calculus.

Upper and lower sums

This access is usually attributed to Jean Gaston Darboux.

The integration interval is thereby broken down into smaller pieces, the searched area decomposes into vertical strips. For each of these strips is now the largest rectangle is considered the one hand, the starting the graph does not intersect the axis (green in the picture), and on the other hand, the smallest rectangle that starting encompassed by the axis of the graph completely (both the green in the picture rectangle with the gray complement it). The sum of the areas of the large rectangles as upper sum, which the small and lower sum. Can you make it through, sufficiently fine subdivision of the integration interval the difference between the upper and lower sum arbitrarily small, so there is only one number that is less than or equal to every upper sum and greater than or equal to every lower sum, and this number is the desired area, the Riemann integral.

For the mathematical clarification be an interval and a bounded function in the following.

Under a decomposition of into parts is understood with a finite sequence. Then the belonging to this disassembly the upper and lower sum are defined as

The function is replaced by the staircase function that the supremum or infimum on each subinterval constant equal to the function on this interval.

In a refinement of the decomposition of the upper sum is smaller, the lower sum is greater (or they stay the same ). So one " infinitely fine " decomposition correspond infimum of the upper sums and supremum of the lower sums; these are referred to as upper or lower darbouxsches integral of:

Each of all possible decompositions of the interval in an arbitrary finite number of subintervals so are considered.

It is always

Applies equality, so called Riemann integrable (or Darboux - integrable ), and the common value

Is called the Riemann integral ( or Darboux integral) of over the interval.

Riemann sums

The above access to the Riemann integral comes via upper and lower sums, as described there, not by Riemann himself but by Jean Gaston Darboux. Riemann studied in the fragmentation of the interval and associated intermediaries sums of the form

Also called Riemann sums or Riemann subtotals relating to the decomposition and the intermediate locations. Riemann called a function over the interval integrable if approaching the Riemann sums with respect to arbitrary decompositions, irrespective of the chosen intermediate points of a fixed number arbitrarily, as long as you only sufficiently fine selects the decompositions. The fineness of a partition Z is thereby the length of the largest subinterval is given by Z, measured, ie by the number:

The number is then the Riemann integral of over. Replacing the illustrations "sufficiently fine" and " approaching any " through precise wording, so can this idea formalize as follows.

A function is called Riemann integrable over the interval when there is a fixed number and to each one, so that for each decomposition with and for any related to intermediaries

Applies. The number is then called the Riemann integral of over and we write it

Riemann integrability

Lebesgue criterion

A function is after the Lebesgue criterion for Riemann integrability - if and only on the compact interval Riemann integrable if it is limited to the interval and almost everywhere continuous. If the function is Riemann - integrable, then it is also Lebesgue integrable and both integrals are identical.

In particular, each control function, any monotonically increasing or monotonically decreasing function and every continuous function is Riemann integrable on a compact Intevall.

Examples

The function with

Is continuous at all irrational numbers and discontinuous at all rational numbers. The set of points of discontinuity is indeed dense in the domain, since this amount but is countable, it is a null set. So this function is Riemann integrable.

The Dirichlet function with

Nowhere is continuous, so it is not Riemann integrable. But it is Lebesgue integrable, as it is almost everywhere zero.

The function with

Has countably many points of discontinuity, so is Riemann integrable. At zero, the right - and left-sided limits do not exist. The function has a discontinuity there is therefore the second kind, the function thus is no control function, that is, it can not be approximated uniformly by step functions. Thus the Riemann integral extends the integral, which is defined by the limit value of step functions of control functions.

Improper Riemann integrals

As an improper Riemann integrals is called:

• Integrals with the interval limits or; is

• Integrals with unbounded functions in the interval boundaries; is

Multidimensional Riemann integral

The multi-dimensional Riemann integral is based on the Jordan measure. Be the n-dimensional Jordan measure and was a jordan- measurable subset. Furthermore, it is a finite sequence of subsets of with and for and should further the function, which returns the maximum distance in a crowd. Set now

Let be a function, ie the sum

Riemann decomposition of the function.

The limit exists

The function is riemann integrable and you set

This integral term has the usual properties of an integral, the integral function is linear and it is Fubini's theorem.

Swell

• Bernhard Riemann: About the representability of a function by a trigonometric series. 1854 ( habilitation thesis with reasons of the integral term named after him ).
• Harro Heuser: Textbook of Analysis 1 9th edition. Teubner, Stuttgart 1991, ISBN 3-519-22231-0 ( in particular section 82).
• Douglas S. Kurtz, Charles W. Swartz: Theories of integration. World Scientific, New Jersey 2004, ISBN 981-256-611-2.