Riemann integral
A generalization of the concept of a Cauchy integral to a certain class of discontinuous functions; introduced by B. Riemann (1853). Consider a function $f$ which is given on an interval $[a,b]$. Let $a=x_0<x_1<\dots<x_n=b$ is a partition (subdivision) of the interval $[a,b]$ and $\Delta x_i = x_i-x_{i-1}$, where $i=1,\dots,n$. The sum
\begin{equation}\label{eq:1}
\sigma = f(\xi_1)\Delta x_1+\dots+f(\xi x_i)\Delta x_i +\dots +f(x_n)\Delta x_n,
\end{equation}
where $x_{i-1}\leq\xi_i\leq x_i$, is called the Riemann sum corresponding to the given partition of $[a,b]$ by the points $x_i$ and to the sample of points $\xi_i$. The number is called the limit of the Riemann sums (1) as
if for any
a
can be found such that
implies the inequality
. If the Riemann sums have a finite limit
as
, then the function
is called Riemann integrable over
, where
. The limit is known as the definite Riemann integral of
over
, and is written as
![]() | (2) |
When then, by definition,
![]() |
and when the integral (2) is defined using the equation
![]() |
A necessary and sufficient condition for the Riemann integrability of over
is the boundedness of
on this interval and the zero value of the Lebesgue measure of the set of all points of discontinuity of
contained in
.
Properties of the Riemann integral.
1) Every Riemann-integrable function on
is also bounded on this interval (the converse is not true: The Dirichlet function is an example of a bounded and non-integrable function on
).
2) The linearity property: For any constants and
, the integrability over
of both functions
and
implies that the function
is integrable over this interval, and the equation
![]() |
holds.
3) The integrability over of both functions
and
implies that their product
is integrable over this interval.
4) Additivity: The integrability of a function over both intervals
and
implies that
is integrable over
, and
![]() |
5) If two functions and
are integrable over
and if
for every
in this interval, then
![]() |
6) The integrability of a function over
implies that the function
is integrable over this interval, and the estimate
![]() |
holds.
7) The mean-value formula: If two real-valued functions and
are integrable over
, if the function
is non-negative or non-positive everywhere on this interval, and if
and
are the least upper and greatest lower bounds of
on
, then a number
can be found,
, such that the formula
![]() | (3) |
holds. If, in addition, is continuous on
, then this interval will contain a point
such that in formula (3),
![]() |
8) The second mean-value formula (Bonnet's formula): If a function is real-valued and integrable over
and if a function
is real-valued and monotone on this interval, then a point
can be found in
such that the formula
![]() |
holds.
References
[1] | B. Riemann, "Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe" H. Weber (ed.) , B. Riemann's Gesammelte Mathematische Werke , Dover, reprint (1953) pp. 227–271 ((Original: Göttinger Akad. Abh. ![]() |
[2] | V.A. Il'in, E.G. Poznyak, "Fundamentals of mathematical analysis" , 1–2 , MIR (1982) (Translated from Russian) |
[3] | L.D. Kudryavtsev, "A course in mathematical analysis" , 1–2 , Moscow (1988) (In Russian) |
[4] | S.M. Nikol'skii, "A course of mathematical analysis" , 1–2 , MIR (1977) (Translated from Russian) |
Comments
References
[a1] | G.E. Shilov, "Mathematical analysis" , 1–2 , M.I.T. (1974) (Translated from Russian) |
[a2] | I.N. Pesin, "Classical and modern integration theories" , Acad. Press (1970) (Translated from Russian) |
[a3] | K.R. Stromberg, "Introduction to classical real analysis" , Wadsworth (1981) |
[a4] | W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1976) pp. 75–78 |
Riemann integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riemann_integral&oldid=29236