翻訳と辞書 Words near each other ・ Rieman Block ・ Riemann (crater) ・ Riemann (surname) ・ Riemann curvature tensor ・ Riemann form ・ Riemann function ・ Riemann hypothesis ・ Riemann integral ・ Riemann invariant ・ Riemann manifold ・ Riemann mapping theorem ・ Riemann problem ・ Riemann series theorem ・ Riemann solver ・ Riemann sphere ・ Riemann sum ・ Riemann surface ・ Riemann Xi function ・ Riemann zeta function ・ Riemann's differential equation ・ Riemann's minimal surface ・ Riemannian ・ Riemannian circle ・ Riemannian connection on a surface ・ Riemannian geometry ・ Riemannian manifold ・ Riemannian Penrose inequality ・ Riemannian submanifold ・ Riemannian submersion ・ Riemannian theory Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Riemann sum ： ウィキペディア英語版 Riemann sum In mathematics, a Riemann sum is an approximation that takes the form $\backslash sum\; f(x)\; \backslash Delta\; x$. It is named after German mathematician Bernhard Riemann. One very common application is approximating the area of functions or lines on a graph, but also the length of curves and other approximations. The sum is calculated by dividing the region up into shapes (rectangles, trapezoids, parabolas, or cubics) that together form a region that is similar to the region being measured, then calculating the area for each of these shapes, and finally adding all of these small areas together. This approach can be used to find a numerical approximation for a definite integral even if the fundamental theorem of calculus does not make it easy to find a closed-form solution. Because the region filled by the small shapes is usually not exactly the same shape as the region being measured, the Riemann sum will differ from the area being measured. This error can be reduced by dividing up the region more finely, using smaller and smaller shapes. As the shapes get smaller and smaller, the sum approaches the Riemann integral. ==Definition== Let ''f'' : ''D'' → R be a function defined on a subset, ''D'', of the real line, R. Let ''I'' = (''b'' ) be a closed interval contained in ''D'', and let :$P=\; \backslash left\; \backslash "\; TITLE="x\_,x\_">)\; \backslash right\; \backslash \},$ be a partition of ''I'', where : A Riemann sum of ''f'' over ''I'' with partition ''P'' is defined as :$S\; =\; \backslash sum\_^\; f(x\_i^$ *)(x_-x_), \quad x_\le x_i^ * \le x_i. Notice the use of "a" instead of "the" in the previous sentence. This is due to the fact that the choice of $x\_i^$ * in the interval $()$ is arbitrary, so for any given function ''f'' defined on an interval ''I'' and a fixed partition ''P'', one might produce different Riemann sums depending on which $x\_i^$ * is chosen, as long as $x\_\backslash le\; x\_i^$ * \le x_i holds true. Example: Specific choices of $x\_i^$ * give us different types of Riemann sums: * If $x\_i^$ *=x_ for all ''i'', then ''S'' is called a left Riemann sum. * If $x\_i^$ *=x_i for all ''i'', then ''S'' is called a right Riemann sum. * If $x\_i^$ *=\tfrac(x_i+x_) for all ''i'', then ''S'' is called a middle Riemann sum. * The average of the left and right Riemann sum is the trapezoidal sum. * If it is given that ::$S\; =\; \backslash sum\_^\; v\_i(x\_-x\_),$ :where $v\_i$ is the supremum of ''f'' over $()$, then ''S'' is defined to be an upper Riemann sum. * Similarly, if $v\_i$ is the infimum of ''f'' over $()$, then ''S'' is a lower Riemann sum. Any Riemann sum on a given partition (that is, for any choice of $x\_i^$ * between $x\_$ and $x\_i$) is contained between the lower and the upper Riemann sums. A function is defined to be ''Riemann integrable'' if the lower and upper Riemann sums get ever closer as the partition gets finer and finer. This fact can also be used for numerical integration. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Riemann sum」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.