Representation

The process of Riemann Integration which is taught in Real Analysis classes is a specific case of the Riemann-Stieltjes Integration. Thus many of the terms and properties used to describe Riemann Integration are discussed in this project and they are extended to the Riemann-Stieltjes integral. This project therefore provides a careful introduction to the theory of Riemann-Stieltjes integration, and explains the properties of this integral. After doing so, we present some applications in functional analysis, where we used the fact that continuous functions on a closed interval are Riemann-Stieltjes Integrable with respect to any function of bounded variation, and this was used in proving the Riesz Representation Theorem. To show versatility of the Riemann-Stieltjes Integral, we also present some applications in Probability Theory, where the integral generates a formula for the Expectation, regardless of its underlying distribution. Other applications considered are population growth, and Mechanics.

HAL (Le Centre pour la Communication Scientifique Directe), 2020

The purpose of this research is to identify students' interpretations when solving Riemann integral problems. Thirteen students enrolled in public university (first year of preparatory class) participated in this study. Data was collected from a test that was proposed at the end of integration courses (Semester II). Through detailed analyses, large majority of students consider the Riemann integral as representing area under a curve or the values of an anti-derivative. In the other side, a few number of students use the limit of approximation conception in their responses. However, understanding the integral as a Riemann sum is highly productive for conceptual learning Tall (1992). Keywords: Teaching and learning of analysis and calculus, teaching and learning of specific topics in university mathematics, Riemann integral.

We consider an extension of the ordinary derivative called the Ω-derivative, and develop some of its properties. Our main result is a generalization of the fundamental Theorem of Calculus that applies to Riemann-Stieltjes integrals in which the integrator is continuous and strictly increasing. Key words and phrases. Riemann-Stieltjes integrals, Fundamental Theorem of Calculus. 2000 AMS Mathematics Subject Classification. 26A06 Resumen. Consideramos una extensión de la derivada usual, llamada la Ω-derivada y desarrollamos algunas de sus propiedades. Nuestro resultado principal es una generalización del teorema fundamental del cálculo que es aplicable a integrales de Riemann-Stieltjes cuyos integradores son continuos y estrictamente crecientes.

SSRN Electronic Journal, 2018

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Comments on the Difficulty and Validity of Various Approaches to the Calculus DAVIDTALL With the introduction of new infinitesimal methods in the last two decades, there are now available a number of dif ferent approaches to the calculus. In her perceptive review essay on "Infinitesimal Calculus" [3]. Peggi Marchi raised For the Learning of Mathematics 2, 2 (November 1981

K. Lee. Lerner. "The Elaboration of the Calculus." (Preprint) Originally published in Schlager, N. Science and Its Times: Understanding the Social Significance of Scientific Discovery. Thomson Gale, 2001

Many of the most influential advances in mathematics during the 18th century involved the elaboration of the calculus, a branch of mathematical analysis which describes properties of functions (curves) associated with a limit process. Although the evolution of the techniques included in the calculus spanned the history of mathematics, calculus was formally developed during the last decades of the 17th century by English mathematician and physicist Sir Isaac Newton (1643-1727) and, independently, by German mathematician Gottfried Wilhelm von Leibniz (1646-1716). Although the logical underpinnings of calculus were hotly debated, the techniques of calculus were immediately applied to a variety of problems in physics, astronomy, and engineering. By the end of the 18th century, calculus had proved a powerful tool that allowed mathematicians and scientists to construct accurate mathematical models of physical phenomena ranging from orbital mechanics to particle dynamics. Although it is clear that Newton made his discoveries regarding calculus years before Leibniz, most historians of mathematics assert that Leibniz independently developed the techniques, symbolism, and nomenclature reflected in his preemptory publications of the calculus in 1684 and 1686. The controversy regarding credit for the origin of calculus quickly became more than a simple dispute between mathematicians. Supporters of Newton and Leibniz often arguing along bitter and blatantly nationalistic lines and the feud itself had a profound influence on the subsequent development of calculus and other branches of mathematical analysis in England and in Continental Europe.

International Journal of Mathematical Education in Science and Technology, 2019

A simple in-class demonstration of integral Calculus for first-time students is described for straightforward whole number area magnitudes, for ease of understanding. Following the Second Fundamental Theorem of the Calculus, macroscopic differences in ordinal values of several integrals, F(x), are compared to the regions of area traced out from the horizontal axis by the derivative functions f (x) over various domains. In addition, microscopic incremental differentials of an integral at a particular position, dF(x), are compared to corresponding values of the derivative function f (x) multiplied by various horizontal shifts, dx. For any area to exist for a derivative function, dx > 0, but the difference between these compared magnitudes collapses to zero as long as dx widths are small. The demonstration readily confirms, both arithmetically and graphically for trigonometric, polynomial, and transcendental functions, the Newton discoveries that (1) the rate that area accumulates under a function is proportional to the ordinal value of the function itself, and (2) changes in elevation along an integral function automatically equal the exact net area traced out by its derivative from the X-axis.

2000

A generalisation of the trapezoid rule for the Riemann-Stieltjes integral and applications for special means are given.

2002

T he fall semester of 1992 was a promising one for me. I had just returned from an extended sabbatical at the University of Utah, I was going to be considered for tenure, and I would be teaching the beginning graduate course in analysis. This article tells the story of how this promise was fulfilled, but in very unexpected ways. One of the analysis students (George Boros), older than the rest and for many years a part-time instructor in the area, had finally decided to pursue a Ph.D. in mathematics. He was well known in New Orleans mathematical circles as “the person who can compute any integral.” Having spent my graduate-student years at Courant Institute, I was comfortably unaware of integrals and did not think that this could be serious mathematics. I was wrong. At the end of the academic year this student asked me to be his adviser. I agreed but cautioned: “George, nobody is going to give you a doctorate in mathematics for computing integrals.” His response was that he was aware...

In Kaur, B., Ho, W.K., Toh, T.L., & Choy, B.H. (Eds.). Proceedings of the 41st Conference of the International Group for the Psychology of Mathematics Education, Vol. 3, pp. 225-232. Singapore: PME., 2017

Action-Process-Object-Schema Theory (APOS) is used to study students' geometric understanding of partition of a rectangular domain and corresponding Riemann sum of an integral of a function of two variables. In this paper we mainly consider the most basic case of a partition, that consisting of a single rectangle (the domain itself). Semi-structured interviews were conducted with ten students who had just finished taking a traditional course in multivariable calculus. Results show that these students had many difficulties with even the most basic mental constructions needed to relate Riemann sum and double integral. This is an important observation since some of these mental constructions are commonly assumed to be obvious to students.