THE DEFINITION OF THE DEFINITE INTEGRAL IN TERMS OF DARBOUX SUMS AND THE CONDITIONS FOR INTEGRABILITY
Abstract
This paper presents a rigorous development of the definite integral through the framework of Darboux sums, emphasizing both theoretical foundations and practical implications. The concepts of upper and lower Darboux sums are introduced as fundamental tools for approximating the area under a bounded function over a closed interval. By examining partitions of the interval and the behavior of supremum and infimum values on subintervals, the study establishes the formal definition of the definite integral. A key focus is placed on the necessary and sufficient conditions for integrability, particularly the convergence of upper and lower sums to a common limit.
Keywords
Definite Integral, Darboux Sums, Upper Sum, Lower Sum, Integrability, Partition of an Interval, Supremum and Infimum, Bounded Functions, Riemann Integral, Real Analysis.
References
- Буваев Қ. Т., Жўраев Ш. Ш. Уч каррали Фурье қаторининг октаэдрли қисмий йиғиндилар кетма-кетлиги учун Лебег ўзгармасини асимптотик ҳолати ҳақида INNOVATION IN THE MODERN EDUCATION SYSTEM jurn.2023,171-172-betlar.
- Буваев Қ. Т., Жўраев Ш. Ш. “Уч каррали Фурье қаторининг октаэдрли қисмий йиғиндилар кетма-кетлигини текис яқинлашиши” ZAMONAVIY MATEMATIKANING DOLZARB MUAMMOLARI VA TATBIQLARI yosh olimlarning ilmiy konferensiyasi tezislari to‘plami.14-15 Mart 2024Toshkent54-56 betlar
- Jo‘rayev S. S. (2026). ELLIPTIK DIFFERENSIAL OPERATORLAR CHIZIQLI SPEKTRAL YOYILMALARINING RAQAMLI IQTISODIYOTDA TUTGAN O‘RNI. Development Of Science, 1(1), pp. 229-236. .
- Jo‘rayev S. S. (2025). MATEMATIKA FANINING RIVOJLANISHI ORQALI INSONLARNING KOINOT HAQIDAGI QARASHLARI. Development Of Science, 12(8), pp. 210-219.
- Jurayev Shaxzod Shuxratjonvich, ENHANCING THE EFFECTIVENESS OF TEACHING METHODOLOGY FOR ELLIPTIC DIFFERENTIAL EQUATION SOLUTIONS BASED ON ARTIFICIAL INTELLIGENCE. (2025). Journal of Multidisciplinary Sciences and Innovations, 4(11), 1764-1768.
- Jurayev Shaxzod Shuxratjonvich, THE ROLE OF THE LAPLACE OPERATOR IN ARTIFICIAL INTELLIGENCE , Journal of Multidisciplinary Sciences and Innovations: Vol. 4 No. 10 (2025): Journal of Multidisciplinary Sciences and Innovations.408-411
- Jurayev Shaxzod Shuxratjonvich ,THE SIGNIFICANCE OF ELLIPTIC DIFFERENTIAL OPERATORS IN ARTIFICIAL INTELLIGENCE SYSTEMS,Journal of Applied SCIENCE AND SOCIAL SCIENCE.650-654
- Behrndt, J., Grubb, G., Langer, M., Lotoreichik, V. (2015). Spectral asymptotics for resolvent differences of elliptic operators with δ and δ′interactions on hypersurfaces. Journal of Spectral Theory..
- Behrndt, J., & Rohleder, J. (2015). Spectral analysis of selfadjoint elliptic differential operators via Dirichlet-to-Neumann maps and abstract Weyl functions. Advances in Mathematics, 274, 242283.
- Shubin, M. A. (1992). Spectral theory of elliptic operators on noncompact manifolds. Astérisque, 207, 35108.
- Алимов Ш. А. Равномерная сходимость и суммируемость спектральных разложений функций из //Дифференц. уравнения.— 1973.— 9, № 4.— С. 669—681.