Engineers and scientists adopt numerical integration to achieve an approximate solution for definite integrals that have no analytic solution. This research focuses on developing some new derivative based quadrature schemes for numerically integrating the integral of Riemann-Stieltjes (Rs-integral) by proposing new schemes which are based on trapezoid-type quadrature. The process of undetermined coefficients has been used for the derivation of proposed schemes. The theoretical derivation and numerical verification of orders of accuracy have been addressed in line with the degrees of precision, and a sufficient improvement has been demonstrated over the existing schemes. The theorems regarding single and mul-tiple use of the suggested schemes in a finite interval have been proved along with the theoretical results on residual terms, both locally and globally. All suggested schemes have been verified to reduce to corresponding variants for the Riemann integral in the case when integrator g(t) = t. Numerical experiments have been performed on all the discussed schemes with the help of MATLAB coding. The experimental results assure the smaller numerical errors by the proposed schemes in the comparison of existing schemes. The obtained results show the efficiency of proposed schemes in light of computational burden and CPU time (in seconds) with reference to the existing quadrature. The proposed work substantially advances the existing knowledge with an addition of derivative-based correction terms in the usual quadrature in a way that the consequent computational costs and execution times are also minimized.

This study investigates the approximation of the natural logarithm function ln(x) for integer values x=2,3,…,100 using two approaches: an infinite geometric series and a Riemann integral–based method developed in this research. The study first proves the recursive formula of ln(x) through the Riemann integral using mathematical induction, establishing a theoretical foundation for the method. Numerical evaluations are then carried out, with the Riemann integral computed both analytically and numerically using partition values n = 10,100,1000,10000,100000,1000000. The two approaches are compared in terms of accuracy and computational efficiency, with accuracy measured using the Mean Absolute Percentage Error (MAPE). The results show that while the geometric series provides higher accuracy, the proposed Riemann integral method demonstrates superior execution speed and serves as a fundamental basis for developing more advanced numerical integration techniques.

This article explores the concept of \emph{absoluteness} in the context of mathematical analysis, focusing specifically on the Riemann integral on $\mathbb{R}^{n}$. In mathematical logic, \emph{absoluteness} refers to the invariance of the truth value of certain statements in different mathematical universes. Leveraging this idea, we investigate the conditions under which the Riemann integral on $\mathbb{R}^{n}$ remains absolute between transitive models of $\mathrm{ZFC}$—the standard axiomatic system in which current mathematics is usually formalized. To this end, we develop a framework for integration on Boolean algebras with respect to finitely additive measures and show that the classical Riemann integral is a particular case of this generalized approach. Our main result establishes that the Riemann integral over rectangles in $\mathbb{R}^{n}$ is absolute in the following sense: if $M \subseteq N$ are transitive models of $\mathrm{ZFC}$, $a, b \in \mathbb{R}^{n} \cap M$, and $f \colon [a, b] \to \mathbb{R}$ is a bounded function in $M$, then $f$ is Riemann integrable in $M$ if, and only if, in $N$ there exists some Riemann integrable function $g \colon [a, b] \to \mathbb{R}$ extending $f$. In this case, the values of the integrals computed in each model are the same. Furthermore, the function $g$ is unique except for a measure zero set.

Fractional fuzzy sets are widely used in decision-making problems; however, they often face limitations in accurately modeling membership and non-membership values. These sets are considered classical models because they use only specific values from the closed interval [0, 1], which leads to a rigid decision-making structure. To address these challenges, we introduce the concept of fractional continuous fuzzy sets, which integrate continuity into the fuzzy framework and use continuous functions instead of fixed numbers. This allows for a broader and more flexible neighborhood around each decision point. The enhanced structure enables the evaluation of both individual points and their surrounding behavior, improving the sensitivity and precision of the decision-making process. We define multiple types of score and accuracy functions based on the Riemann integral, and utilize function-theoretic tools to analyze the continuous fuzzy data. Two decision-making algorithms are proposed and applied to a real-world industrial robot selection problem. The results are then compared with existing MCDM methods, demonstrating the effectiveness, flexibility, and reliability of the proposed approach.

The natural logarithm (ln) function plays a critical role in higher education, particularly in equipping students with advanced problem-solving skills applicable across science, technology, engineering, and mathematics (STEM) disciplines. This study has two primary objectives: first, to explore and compare the derivation of the natural logarithm using the Riemann integral and geometric series methods; and second, to examine the pedagogical implications of these approaches by analyzing student perceptions. Data were collected from 23 students using a questionnaire comprising five items, each scored on a scale of 1 to 10, to evaluate understanding and perception of both methods. Results from a paired t-test indicate that the Riemann integral method is considered superior to the geometric series in terms of conceptual understanding of the ln function (p < 0.001), ease of memorization (p < 0.001), manual computation (p = 0.032), and implementation in computer programming (p < 0.001). However, no significant difference was found between the two methods regarding the perceived difficulty of ln calculation (p = 0.660). Notably, the geometric series was favored for manual computations due to its simplicity

Neutrosophic Set Theory (NST) is an extension of Intuitionistic Fuzzy Set Theory (IFST). While IFST relies on two possibilities for the complete depiction of a set, neutrosophic set theory familiarizes an additional third possibility, thus providing a more delicate representation. Our research builds upon a further extension of neutrosophic set theory, known as quadri-partitioned neutrosophic set theory (QPNST), which brings in a fourth possibility for a more detailed and complete description of sets. In this study, we define the Riemann Integral Theory (RIT) within the framework of QPNST. This opens new doors for probing the properties and characteristics of the Riemann integral in this extended context. One strategic concept that arises in this work is the level cut. In QPNST, the level cut is defined as a four-tuple (i, j,k, l), which represents the different possibilities inherent in the theory. The notion of the Quadri-Partitioned Neutrosophic Riemann Integral Theory (QPNRIT) is explored numerically inthis study, and the results are systematically presented in tabular form. This numerical approach sheds light on the integral’s properties and facilitates the understanding of its behavior within the QPNST framework. This study explores quadripartitioned neutrosophic soft topological spaces, extending neutrosophic set theory (NST), which incorporates three membership values: true, false, and indeterminacy. The study introduces new concepts such as QPNS semi-open, QPNS pre-open, and QPNS ∗b open sets, and builds on these to define QPNS closure, exterior, boundary, and interior. A key development is the definition of a quadripartitioned neutrosophic soft base, which plays a central role in these topological structures. The paper also explores the concept of a quadripartitioned neutrosophic soft sub-base and discusses local bases, as well as the first- and second-countability axioms. The study further examines hereditary properties of these spaces, distinguishing between inherited and non-inherited properties. Key results include that a quadripartitioned neutrosophic soft subspace of a first-countable space is also first-countable, and a second-countable subspace of a second-countable space remains second-countable. It also highlights the relationship between second countability and separability in these spaces, asserting that a second-countable quadripartitioned neutrosophic soft space is separable, though the converse is not always true. This work lays the foundation for further research in neutrosophic soft topologies.

Abstract Large deflection analysis is crucial for understanding and predicting the mechanical performance of flexible beam structures, which can be used to analyze metamaterial unit cells simplified into flexible beam structures. This paper investigates the large deflection behavior of an inclined cantilever beam with its freedom end subjected to a dead load. Firstly, considering the geometric nonlinearity of the beam and the influence of boundary conditions, a mathematical model of the beam is established and solved. Secondly, equations for the deflection curve and strain energy are derived, expressed in the semi-analytical form of elliptic functions. Then, a program is developed using Riemann integration combined with the bisection method to iteratively obtain the final calculation results. Finally, the calculation results of this paper are compared with those obtained by the nonlinear finite element method, thereby validating the accuracy of the proposed algorithm.

Henstock integral is a generalized version of the Riemann integral and in most cases, it is more general than the Lebesgue integral that is not constructed through a measure theoretic standpoint. The PU integral, on one hand, is a Henstock type that utilizes the notion of a partition of unity. In this paper, the Saks-Henstock Lemma and the Change of variable formula for the PU integral will be established.

In this paper, we analyze an optimal control problem with a mixed cost function, which combines a terminal cost at the final state and an integral term involving the state and control variables. The problem includes both state and control constraints, which adds complexity to the analysis. We establish a necessary optimality condition in the form of the maximum principle, where the adjoint equation is an integral equation involving the Riemann and Stieltjes integrals with respect to a Borel measure. Our approach is based on the Dubovitskii–Milyutin theory, which employs conic approximations to efficiently manage state constraints. To illustrate the applicability of our results, we consider two examples related to epidemiological models, specifically the SIR model. These examples demonstrate how the developed framework can inform optimal control strategies to mitigate disease spread. Furthermore, we explore the implications of our findings in broader contexts, emphasizing how mixed cost functions manifest in various applied settings. Incorporating state constraints requires advanced mathematical techniques, and our approach provides a structured way to address them. The integral nature of the adjoint equation highlights the role of measure-theoretic tools in optimal control. Through our examples, we demonstrate practical applications of the proposed methodology, reinforcing its usefulness in real-life situations. By extending the Dubovitskii–Milyutin framework, we contribute to a deeper understanding of constrained control problems and their solutions.

Among several classes of Riemann integrable real valued functions we are interested in finding class of functions whose (signed) integral values over a compact interval vanishes. Determining such class of functions was one of the important objectives of this paper. We begin our quest by introducing Bernoulli numbers then extending them to Bernoulli polynomials. We observe that Bernoulli polynomials are generalized version of the most famous and notorious Bernoulli numbers introduced by Jacob Bernoulli in 1713. In particular, we see that Bernoulli numbers are simply the constant terms of Bernoulli polynomials. Bernoulli numbers and Bernoulli polynomials play very big role in analyzing several aspects of mathematics and they occur unexpectedly in several counting problems. We prove some interesting properties of Bernoulli polynomials which generate another class of functions having the property of zero area in [0,1]. In this paper, we try to establish that such class of functions are precisely the Bernoulli polynomials and prove that the Riemann integral of five categories of Bernoulli polynomials over the compact interval [0,1] is zero. The geometric meaning of this fact for special cases is explained through several figures which will provide better insight and understanding. This paper will also provide an scope for generalizing in the analysis of Riemann integration of Bernoulli polynomials not restricted to just the interval [0,1] but for any compact interval in the real line.

Abstract Continuity is one of the most central notions in mathematics, physics and computer science. An interesting associated topic is decompositions of continuity, where continuity is shown to be equivalent to the combination of two or more weak continuity notions. In this paper, we study the logical properties of basic theorems about weakly continuous functions, like the supremum principle for the unit interval. We establish that most weak continuity notions are as tame as continuity, that is, the supremum principle can be proved from the relatively weak arithmetical comprehension axiom only. By contrast, for seven ‘wild’ weak continuity notions, the associated supremum principle yields rather strong axioms, including Feferman's projection principle, full second‐order arithmetic or Kleene's associated quantifier . Working in Kohlenbach's higher‐order Reverse Mathematics, we also obtain elegant equivalences in various cases and obtain similar results for for example, Riemann integration. We believe these results to be of interest to mainstream mathematics as they cast new light on the distinction of ‘ordinary mathematics’ versus ‘foundations of mathematics/set theory’.

The aim of this paper to presented a class of interesting fuzzy integral nonlinear equations involving nonlinear fuzzy function of other multi fuzzy nonlinear functions that presented in fist time, the formula under study is an interesting type of a fuzzy nonlinear Volterra integral equation which defined on fuzzy numbers that has lower and upper functions. The Housdorf fuzzy metric space on fuzzy numbers and a fuzzy metric space on a fuzzy continuous functions and the space of all uniform modulus of absolutely fuzzy continuous functions have been given in details. Also the observer results investigated on parameter cases of fuzzy problem formulation, some necessary and sufficient conditions of boundedness and locally Lipschitz for different fuzzy functions and differential of them and also using Leibnitz rule for presented fuzzy problem formulation. The locally existence and uniqueness solutions in the space of all uniform modulus of absolutely fuzzy continuous functions which is generalized of fuzzy number space and established on contraction condition which proved by using the properties of fuzzy metric space this certain space. Also used concept of fuzzy Riemann integral and normed function to prove interesting results and illustrated in some examples to clear the proposal fuzzy integral nonlinear equations has a solution in certain fuzzy space that in firstly explained with different metrics spaces and normed properties and satisfied all the hypothesis for the component of the equation and obtain the estimations of the suggested conditions on certain fuzzy metrics as a stuffiest condition for grantee the assuming equations has a fuzzy solution with their lower and upper parameters.

This paper introduces the notion of Riemann integration for quotientspace valued functions on time scales. Quotient Riemann ∆-integral, quotient Riemann ∇-integral and quotient Riemann ♢α-integral are defined.Results establishing that ∆- and ∇-integrals are special cases of quotientRiemann ♢α-integral are observed; and a few standard results formulated. The relation between Banach valued Riemann integral and quotient valued Riemann integral is established. Notion of continuity of functions usingquotient norm is defined and it’s integrability proved.

This paper develops a novel Milne inequality for third-differentiable and h-convex functions using Riemann integrals. Furthermore, new Milne inequalities are proposed utilizing a summation parameter p≥1 for s-convexity, convexity, and P-functions class. We examine cases when the third derivative functions are also bounded and Lipschitzian.

The Jackson integral represents a q-analogue of the Riemann integral, thereby extending the integration concept into the domain of q-calculus, while the Riemann integral remains a traditional calculus tool for assessing the area under a curve. It is well known that Henstock-Kurzweil integral is a generalized Riemann integral. In this article, we introduce q-analogue of Henstock-Kurzweil integral, called q-Henstock-Kurzweil integral. We discuss several important properties of newly introduce q-Henstock-Kurzweil integrals and its some results. Moreover, we show that q-Henstock-Kurzweil integrable functions contain Henstock-Kurzweil integrable functions. Furthermore, we introduce Fundamental Theorem of Calculus for q-Henstock-Kurzweil integrable functions in q-analogous approach. Finally, using this integrable functions we suggest a solution method for a class of linear fractional q-differential equations.

This paper develops a novel Bullen inequality for third-differentiable functions using Riemann integrals. Furthermore, new Bullen inequalities are proposed utilizing a summation parameter p ? 1 for and s-convex functions,convex functions and P-functions classes. Particular cases are studied when the third derivative functions are also bounded and Lipschitzian.

This paper deals with a new sharp version of Simpson’s second inequality by using the concepts of absolute continuity, Grüss inequality, and Chebyshev functionals. To demonstrate the applicability of the main result, three examples are given. Also, as generalization of the main result, a Simpson’s second type inequality related to the class of Riemann–Liouville fractional integrals is obtained. In addition, Simpson’s 3/8 formula is applied to approximate the Riemann integral of an absolutely continuous function as well as estimation of approximation error.

In this paper, we establish a stochastic integral, relying on the order-complete vector lattices’ properties. This stochastic integral is alike to the Riemann integral on bounded real-valued functions. This integral’s properties provide stochastic integration of stochastic processes beyond integral, with respect to some Brownian motion. The Arrhenius equation, which is actually an essential equation in Physical Chemistry, does provide a capital transfer model between two assets in a specific portfolio. The Arrhenius equation is actually modified under this stochastic integral. The stochastic term of such a stochastic process is alike to the introduction of the “white noise” in stochastic integration, with respect to some Brownian motion.

In Integral Calculus the classic problem is the determination of the area under the curve, when said region is not expressible in terms of elementary figures. This translates into a multiplicity of problems and exercises that are presented to students in a Calculus course. This article presents a useful problem for Mathematics Education, derived from a generalized integral operator, for this we define what we understand by an integrable function in this generalized sense, and the geometric interpretation of a generalized definite integral is presented. The interesting thing about this generalization is that said geometric interpretation is similar to the geometric interpretation of the classical Riemann integral, but not in the xy plane, but in the Ty plane, where T is the kernel of the generalized integral.

Algebrability and Riemann integrability of the composite function