Riemann integral

The Riemann integral (also Riemann integral ) is a method named after the German mathematician Bernhard Riemann for specifying the descriptive representation of the area between the -axis and the graph of a function . The Riemann integral concept belongs to the two classical ones of analysis alongside the more general Lebesgue one. In many applications only integrals of continuous or piecewise continuous functions are required. Then the somewhat simpler but less general concept of the integral of control functions is sufficient . ${\displaystyle x}$

The concept on which the Riemann integral is based consists in approximating the area sought with the aid of the area of ​​rectangles, which is easy to calculate. The procedure is to choose two families of rectangles in each step so that the graph of the function lies “between” them. By successively increasing the number of rectangles, one obtains an increasingly more precise approximation of the function graph over time through the step functions belonging to the rectangles . Accordingly, the area between the graph and the axis can be approximated by the area of ​​the rectangles . ${\displaystyle x}$

Definitions

There are essentially two common methods for defining the Riemann integral:

• the method ascribed to Jean Gaston Darboux using upper and lower sums and
• Riemann's original method using Riemann sums.

The two definitions are equivalent: every function can be integrated in the Darbouxian sense if and only if it can be integrated in the Riemannian sense; in this case the values ​​of the two integrals agree. In typical introductions to analysis, especially in schools, Darboux's formulation is largely used for definition. Riemann sums are often added as a further aid, for example to prove the main theorem of integral and differential calculus .

Upper and lower sums

This access is mostly attributed to Jean Gaston Darboux .

Lower sum (green) and upper sum (green plus lavender) for a division into four sub-intervals

The integration interval is broken down into smaller pieces, and the area you are looking for is broken down into vertical strips. For each of these stripes, on the one hand the largest rectangle that does not intersect the graph starting from the -axis (green in the picture), and on the other hand the smallest rectangle that completely surrounds the graph starting from the -axis (in the picture the green Rectangle together with the gray addition above). The sum of the areas of the large rectangles is called the upper sum , that of the small ones is called the lower sum . If one can make the difference between the upper and lower sums as small as desired by means of a suitable, sufficiently fine subdivision of the integration interval, then there is only one number that is less than or equal to each upper sum and greater than or equal to each lower sum, and this number is the area of ​​the search, the Riemann integral. ${\displaystyle x}$ ${\displaystyle x}$

For the mathematical precision in the following an interval and a restricted function are assumed . ${\displaystyle [a,b]}$ ${\displaystyle f\colon [a,b]\to \mathbb {R} }$

A division of into parts means a finite sequence with . Then the upper and lower sums belonging to this decomposition are defined as ${\displaystyle Z}$ ${\displaystyle [a,b]}$ ${\displaystyle n}$ ${\displaystyle x_{0},x_{1},\dotsc ,x_{n}}$ ${\displaystyle a=x_{0}<x_{1}<\dotsb <x_{n}=b}$

${\displaystyle O(Z):=\sum _{k=1}^{n}{\Big (}(x_{k}-x_{k-1})\cdot \sup _{x_{k-1}<x<x_{k}}f(x){\Big )}}$

${\displaystyle U(Z):=\sum _{k=1}^{n}{\Big (}(x_{k}-x_{k-1})\cdot \inf _{x_{k-1}<x<x_{k}}f(x){\Big )}}$.

The function is replaced by the step function , which is constantly equal to the supremum or infimum of the function on this interval on each sub- interval.

With a finer subdivision, the upper total becomes smaller and the lower total becomes larger

If the decomposition is refined, the upper sum becomes smaller and the lower sum larger (or they remain the same). An “infinitely fine” decomposition therefore corresponds to the infimum of the upper sums and the supremum of the lower sums; these are called the upper or lower Darboux integral of : ${\displaystyle f}$

${\displaystyle {\overline {\int _{a}^{b}}}f(x)\,\mathrm {d} x:=\inf _{Z}O(Z):=\inf\{O(Z):Z{\mbox{ ist Zerlegung von }}[a,b]\}}$

${\displaystyle {\underline {\int _{a}^{b}}}f(x)\,\mathrm {d} x:=\sup _{Z}U(Z):=\sup\{U(Z):Z{\mbox{ ist Zerlegung von }}[a,b]\}}$.

All possible breakdowns of the interval into any finite number of sub-intervals are therefore considered. ${\displaystyle Z}$

Example of dividing an interval [a, b] into n = 8 parts (upper sum purple and lower sum orange)

It always applies

${\displaystyle {\underline {\int _{a}^{b}}}f(x)\,\mathrm {d} x\leq {\overline {\int _{a}^{b}}}f(x)\,\mathrm {d} x.}$

If equality applies, then Riemann-integrable (or Darboux-integrable ), and the common value ${\displaystyle f}$

${\displaystyle \int _{a}^{b}f(x)\,\mathrm {d} x:={\underline {\int _{a}^{b}}}f(x)\,\mathrm {d} x={\overline {\int _{a}^{b}}}f(x)\,\mathrm {d} x}$

is called the Riemann integral (or Darboux integral ) of over the interval . ${\displaystyle f}$ ${\displaystyle [a,b]}$

Riemann sums

The above access to the Riemann integral via upper and lower sums, as described there, does not come from Riemann himself, but from Jean Gaston Darboux . Riemann examined sums of the form for a decomposition of the interval and the corresponding intermediate points ${\displaystyle Z=\{a=x_{0},x_{1},\dotsc ,x_{n}=b\}}$ ${\displaystyle [a,b]}$ ${\displaystyle Z}$ ${\displaystyle t_{i}\in [x_{i-1},x_{i}](i=1,\dotsc ,n)}$

${\displaystyle S(f,Z,t_{1},\dotsc ,t_{n})=\sum _{i=1}^{n}f(t_{i})(x_{i}-x_{i-1}),}$

Geometric illustration of the Riemann subtotals (orange rectangles). It applies to the disassembly shown ${\displaystyle Z\colon \mu (Z)=x_{7}-x_{6}}$

also referred to as Riemann sums or Riemann subtotals with regard to the decomposition and the intermediate points. Riemann called a function integrable over the interval if the Riemann sums approach a fixed number with respect to any decomposition regardless of the selected intermediate positions , provided that the decomposition is chosen with sufficient precision. The fineness of a decomposition Z is measured over the length of the largest sub-interval given by Z , i.e. by the number: ${\displaystyle Z}$ ${\displaystyle t_{1},\dotsc ,t_{n}}$ ${\displaystyle f}$ ${\displaystyle [a,b]}$ ${\displaystyle A}$ ${\displaystyle Z}$ ${\displaystyle [x_{i-1},x_{i}]}$

${\displaystyle \mu (Z)=\max\{x_{i}-x_{i-1}:i=1,\dotsc ,n\}.}$

The number is then the Riemann integral of over . If one replaces the illustrations “sufficiently fine” and “arbitrarily approximated” with a precise formulation, this idea can be formalized as follows. ${\displaystyle A}$ ${\displaystyle f}$ ${\displaystyle [a,b]}$

A function is called over the interval integrated Riemann, if there is a fixed number and each one is so that for each separation, with and for arbitrary to corresponding intermediate locations ${\displaystyle f\colon [a,b]\rightarrow \mathbb {R} }$ ${\displaystyle [a,b]}$ ${\displaystyle A}$ ${\displaystyle \varepsilon >0}$ ${\displaystyle \delta >0}$ ${\displaystyle Z}$ ${\displaystyle \mu (Z)<\delta }$ ${\displaystyle Z}$ ${\displaystyle t_{1},\dotsc ,t_{n}}$

${\displaystyle |S(f,Z,t_{1},\dotsc ,t_{n})-A|<\varepsilon }$

applies. The number is then called the Riemann integral of over and you write for it ${\displaystyle A}$ ${\displaystyle f}$ ${\displaystyle [a,b]}$

${\displaystyle A=\int _{a}^{b}f}$or . ${\displaystyle \displaystyle A=\int _{a}^{b}f(x)\;\mathrm {d} x}$

Riemann integrability

Lebesgue criterion

One on a compact interval bounded function is after the Lebesgue criterion for Riemann integrability exactly then to be integrated Riemann, if on this interval almost anywhere ever is. If the function is Riemann integrable, it is also Lebesgue integrable and both integrals are identical. ${\displaystyle [a,b]}$ ${\displaystyle f}$ ${\displaystyle [a,b]}$

In particular, every control function , every monotonically increasing or monotonously decreasing function and every continuous function can be Riemann-integrated over a compact interval .

Examples

The function with ${\displaystyle f\colon [0,1]\to [0,1]}$

${\displaystyle f(x)={\begin{cases}1&~{\text{falls}}~x=0\\{\frac {1}{q}}&~{\text{falls}}~x={\frac {r}{q}}~{\text{mit}}~r,q\in \mathbb {N} ~{\text{teilerfremd}}\\0&~{\text{falls}}~x\in \mathbb {R} \setminus \mathbb {Q} \end{cases}}}$

is continuous in all irrational numbers and discontinuous in all rational numbers. The set of points of discontinuity is close to the definition range, but since this set is countable , it is a zero set . The function can thus be integrated into Riemann.

The Dirichlet function with ${\displaystyle g\colon [0,1]\to [0,1]}$

${\displaystyle g(x)={\begin{cases}1&~{\text{falls}}~x\in \mathbb {Q} \\0&~{\text{falls}}~x\in \mathbb {R} \setminus \mathbb {Q} \end{cases}}}$

is nowhere continuous, so it is not Riemann integrable. But it can be integrated into Lebesgue because it is almost everywhere zero.

The function with ${\displaystyle h\colon [-1,1]\to [0,1]}$

${\displaystyle h(x)={\begin{cases}1&~{\text{falls}}~x\in \{{\frac {1}{n}}\mid n\in \mathbb {N} \}\\0&~{\text{sonst}}~\end{cases}}}$

has a countable number of discontinuities, so it can be integrated by Riemann. If the value is zero, the right-hand limit does not exist. The function therefore has a point of discontinuity of the second type there. The function is therefore not a control function , that is, it cannot be approximated uniformly by step functions . The Riemann integral extends the integral that is defined via the limit value of step functions of control functions.

Improper Riemann integrals

→ Main article : improper integral

Improper Riemann integrals are called:

• Integrals with the interval limits or ; is there ${\displaystyle -\infty }$ ${\displaystyle \infty }$

${\displaystyle \int _{a}^{\infty }f(x)\,\mathrm {d} x=\lim _{\beta \to \infty }\int _{a}^{\beta }f(x)\,\mathrm {d} x}$,

${\displaystyle \int _{-\infty }^{b}f(x)\,\mathrm {d} x=\lim _{\alpha \to -\infty }\int _{\alpha }^{b}f(x)\,\mathrm {d} x}$ and

${\displaystyle \int _{-\infty }^{\infty }f(x)\,\mathrm {d} x=\int _{-\infty }^{a}f(x)\,\mathrm {d} x+\int _{a}^{\infty }f(x)\,\mathrm {d} x}$ with any ${\displaystyle a\in \mathbb {R} .}$

• Integrals with unbounded functions in one of the interval boundaries; is there

${\displaystyle \int _{a}^{b}f(x)\,\mathrm {d} x=\lim _{\varepsilon \searrow 0}\int _{a+\varepsilon }^{b}f(x)\,\mathrm {d} x}$ or. ${\displaystyle \displaystyle \int _{a}^{b}f(x)\,\mathrm {d} x=\lim _{\varepsilon \searrow 0}\int _{a}^{b-\varepsilon }f(x)\,\mathrm {d} x.}$

Multi-dimensional Riemannian integral

The multi-dimensional Riemann integral is based on the Jordan measure . Let be the n-dimensional Jordan measure and be a Jordan-measurable subset. In addition, let be a finite sequence of subsets of with and for and further be the function that returns the maximum distance in a set . Sit now ${\displaystyle \mu _{n}}$ ${\displaystyle E\subset \mathbb {R} ^{n}}$ ${\displaystyle \tau =(E_{i})_{i=0}^{k}}$ ${\displaystyle E}$ ${\displaystyle \textstyle \bigcup \nolimits _{i=0}^{k}E_{i}=E}$ ${\displaystyle \mu _{n}(E_{i}\cap E_{j})=0}$ ${\displaystyle i\neq j}$ ${\displaystyle \textstyle \operatorname {diam} (A)=\sup _{a,b\in {\overline {A}}}^{\,}(\|a-b\|)}$ ${\displaystyle A}$

${\displaystyle \delta _{\tau }=\max _{i=0,\dotsc ,k}\left(\operatorname {diam} (E_{i})\right)=\max _{i=0,\dotsc ,k}\left(\sup _{a,b\in {\overline {E_{i}}}}^{\,}(\|a-b\|)\right)}$.

Let be a function, then the sum is called ${\displaystyle f\colon E\to \mathbb {R} }$

${\displaystyle \rho _{\tau }(f,\xi ^{(0)},\dotsc ,\xi ^{(k)})=\sum _{i=0}^{k(\delta _{\tau })}f(\xi ^{(i)})\mu _{n}(E_{i})}$

Riemann decomposition of the function . ${\displaystyle f}$

Does the limit exist

${\displaystyle \lim _{\delta _{\tau }\to 0}\rho _{\tau }(f,\xi ^{(0)},\dotsc ,\xi ^{(k)})=\lim _{\delta _{\tau }\to 0}\sum _{i=0}^{k(\delta _{\tau })}f(\xi ^{(i)})\mu _{n}(E_{i})}$,

so the Riemann function can be integrated and one sets ${\displaystyle f}$

${\displaystyle \int _{E}f(x)\mathrm {d} x=\lim _{\delta _{\tau }\to 0}\sum _{i=0}^{k(\delta _{\tau })}f(\xi ^{(i)})\mu _{n}(E_{i})}$.

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

Birkhoff integral

→ Main article : Birkhoff integral

The Birkhoff integral represents a generalization of the Riemann integral for Banach space- valued functions. In particular, this generalizes the approach using Riemann sums.

swell

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

Web links

Wikibooks: Math for non-freaks: Riemannintegral  - learning and teaching materials

• Visualization of the Riemann integral at GeoGebra
• Visualization of the Riemann integral in Visual Calculus
• Visualization of the Riemann integral on math-online
• Multi-dimensional integrals at Springer