Riemann Integration in the Euclidean Space
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909154567782400 |
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| author | Segun, Akerele Olofin |
| author_facet | Segun, Akerele Olofin |
| contents | The so-called Riemann sums have their origin in the efforts of Greek mathematicians to find the center of gravity or the volume of a solid body. These researches led to the method of exhaustion, discovered by Archimedes and described using modern ideas by MacLaurin in his \textit{Treatise of Fluxions} in 1742. At this times the sums were only a practical method for computing an area under a curve, and the existence of this area was considered geometrically obvious. The method of exhaustion consists in almost covering the space enclosed by the curve with $n$ geometric objects with well-known areas such as rectangles or triangles, and finding the limit (though this topic was very blurry at these early times) when $n$ increases. One of its most remarkable application is squaring the area $\mathcal{A}$ enclosed by a parabola and a line. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_19703 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Riemann Integration in the Euclidean Space Segun, Akerele Olofin History and Overview The so-called Riemann sums have their origin in the efforts of Greek mathematicians to find the center of gravity or the volume of a solid body. These researches led to the method of exhaustion, discovered by Archimedes and described using modern ideas by MacLaurin in his \textit{Treatise of Fluxions} in 1742. At this times the sums were only a practical method for computing an area under a curve, and the existence of this area was considered geometrically obvious. The method of exhaustion consists in almost covering the space enclosed by the curve with $n$ geometric objects with well-known areas such as rectangles or triangles, and finding the limit (though this topic was very blurry at these early times) when $n$ increases. One of its most remarkable application is squaring the area $\mathcal{A}$ enclosed by a parabola and a line. |
| title | Riemann Integration in the Euclidean Space |
| topic | History and Overview |
| url | https://arxiv.org/abs/2403.19703 |