{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T00:50:52Z","timestamp":1760057452309,"version":"build-2065373602"},"reference-count":33,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2025,2,4]],"date-time":"2025-02-04T00:00:00Z","timestamp":1738627200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Our goal is to prove the Euler\u2013Maclaurin summation formula using only the Taylor formula. Furthermore, the Euler\u2013Maclaurin summation formula will be considered for the case of functions of the class C2k([a,b]). A stronger version of the estimation of the rest for functions of class Cn([a,b]) is given. We will also present the application of the obtained Euler\u2013Maclaurin summation formulas to determine the asymptotics of Riemann sums for uniform partitions of intervals of integration. This paper also introduces the concepts of the asymptotic smoothing of the graph of a given function, as well as the asymptotic uniform distribution of the positive and negative values of the given function (generalizing the concept of the symmetric distribution of these values with respect to the x-axis, as in the case of the Peano\u2013Jordan measure).<\/jats:p>","DOI":"10.3390\/sym17020226","type":"journal-article","created":{"date-parts":[[2025,2,4]],"date-time":"2025-02-04T08:25:23Z","timestamp":1738657523000},"page":"226","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Asymptotics of Riemann Sums for Uniform Partitions of Intervals of Integration"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8497-8262","authenticated-orcid":false,"given":"Marcin","family":"Adam","sequence":"first","affiliation":[{"name":"Department of Mathematics, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9769-7497","authenticated-orcid":false,"given":"Jakub Jan","family":"Ludew","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2296-265X","authenticated-orcid":false,"given":"Micha\u0142","family":"R\u00f3\u017ca\u0144ski","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0865-3554","authenticated-orcid":false,"given":"Adrian","family":"Smuda","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6503-5675","authenticated-orcid":false,"given":"Roman","family":"Witu\u0142a","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland"}]}],"member":"1968","published-online":{"date-parts":[[2025,2,4]]},"reference":[{"key":"ref_1","unstructured":"Knuth, D. 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