Mo logo [home] [lexicon] [problems] [tests] [courses] [auxiliaries] [notes] [staff] german flag

Mathematics-Online lexicon: Annotation to

Riemann Integral

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z overview

The definite integral of a piecewise continuous function $ f$ is defined by

$\displaystyle \int_a^b f(x)\,dx = \lim_{\vert\Delta\vert\to0} \int_a^b f_\Delta =
\lim_{\vert\Delta\vert\to0} \sum_{k} f(\xi_k)\,\Delta x_k \quad .

Here $ \Delta:\,a=x_0<x_1<\cdots<x_n=b$ is a partition of $ [a,b]$;

$\displaystyle \vert\Delta\vert=\max_k \Delta x_k\,, \qquad \Delta x_k=x_k-x_{k-1}\,, $

denotes the maximal length of the interval and $ \xi_k$ is an arbitrary point in the $ k$-th interval. The sums on the side of the integral's definition are called Riemann sums.


For a positive function $ f$, $ \int_a^b f$ corresponds to the area below the graph of $ f$.

Slightly strengthening the hypotheses, we assume that $ f$ is continuously differentiable, and prove the convergence of the Riemann sums using Cauchy's criterion.

For a sequence $ (\Delta_i)$ of partitions with $ \vert\Delta_i\vert\to 0$, we compare the Riemann sums corresponding to $ \Delta_m$ and $ \Delta_n$ with the aid of a partition $ \Delta$, which consists of the union of the points of $ \Delta_m$ and $ \Delta_n$:

$\displaystyle \Delta_m: x_0<\cdots<x_{k_m},\,
\Delta_n: y_0<\cdots<y_{k_n}
\Delta: z_0<\cdots<z_k

The difference of the Riemann sum for $ \Delta$,

$\displaystyle \sum_{j=1}^k f(\zeta_j)\,\Delta z_j,\quad

to the Riemann sum for $ \Delta_m$ can be estimated via the mean value theorem. From

$\displaystyle \vert f(t_1)-f(t_2)\vert \le (t_2-t_1)
\max_{t\in [t_1,t_2]} \vert f^\prime(t)\vert

we obtain
$\displaystyle \left\vert\int f_\Delta- \int f_{\Delta_m}\right\vert$ $\displaystyle =$ $\displaystyle \left\vert \sum_{j=1}^{k} f(\zeta_j)\,\Delta z_j -
\sum_{i=1}^{k_m} f(\xi_i)\,\Delta x_i\right\vert$  
  $\displaystyle =$ $\displaystyle \left\vert \sum_{i=1}^{k_m}
\sum_{x_{i-1}\le z_{j-1}<z_{j}\le x_{i}}
(f(\zeta_j)-f(\xi_i))\Delta z_j\right\vert$  
  $\displaystyle \leq$ $\displaystyle \vert\Delta_m\vert\,
\underbrace{\max_{t\in[a,b]} \vert f^\prime(...
\sum_{x_{i-1}\le z_{j-1}<z_{j}\le x_{i}}
\Delta z_j


$\displaystyle \left\vert\int f_\Delta- \int f_{\Delta_m}\right\vert
\le c\,(b-a)\,\vert\Delta_m\vert

With an analogous estimate for $ \vert\int f_\Delta-\int f_{\Delta_n}\vert$, Cauchy's criterion follows:

$\displaystyle \left\vert\int f_{\Delta_m}-\int f_{\Delta_n}\right\vert
\le c\,(b-a)\,\left(\vert\Delta_m\vert+\vert\Delta_n\vert\right)\to 0$   for$\displaystyle \quad

The convergence of two sequences to the same limit can be established with an identical argument.

For piecewise continuous $ f$, the proof is technically more complicated. It relies on the uniform continuity of $ f$ on the closed interval of integration:

$\displaystyle \left\vert f(x_1)-f(x_2)\right\vert\leq\varepsilon$   für$\displaystyle \quad
\vert x_1-x_2\vert<\delta


  automatisch erstellt am 22.  9. 2016