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In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval.〔The Riemann integral was introduced in Bernhard Riemann's paper "Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe" (On the representability of a function by a trigonometric series; i.e., when can a function be represented by a trigonometric series). This paper was submitted to the University of Göttingen in 1854 as Riemann's ''Habilitationsschrift'' (qualification to become an instructor). It was published in 1868 in ''Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen'' (Proceedings of the Royal Philosophical Society at Göttingen), vol. 13, pages 87-132. (Available on-line (here ).) For Riemann's definition of his integral, see section 4, "Über der Begriff eines bestimmten Integrals und den Umfang seiner Gültigkeit" (On the concept of a definite integral and the extent of its validity), pages 101-103.〕 For many functions and practical applications, the Riemann integral can be evaluated by the fundamental theorem of calculus or approximated by numerical integration. The Riemann integral is unsuitable for many theoretical purposes. Some of the technical deficiencies in Riemann integration can be remedied with the Riemann–Stieltjes integral, and most disappear with the Lebesgue integral. == Overview == Let be a non-negative real-valued function on the interval , and let : be the region of the plane under the graph of the function and above the interval (see the figure on the top right). We are interested in measuring the area of . Once we have measured it, we will denote the area by: : The basic idea of the Riemann integral is to use very simple approximations for the area of . By taking better and better approximations, we can say that "in the limit" we get exactly the area of under the curve. Note that where can be both positive and negative, the definition of is modified so that the integral corresponds to the ''signed area'' under the graph of : that is, the area above the -axis minus the area below the -axis. 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 integral」の詳細全文を読む スポンサード リンク
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