Thesis (M. Phil.)--Hong Kong University of Science and Technology, 2002. / Includes bibliographical references (leaves 40). Also available in electronic version. Access restricted to campus users.

The principal purpose of this paper is to develop the properties of Riemann-Stieljes Integrals. Consequently, a knowledge of the terminology and theory of point sets and functions of real variables shall be assumed.

This paper is a study of the Stieltjes integral, a generalization of the Riemann integral normally studied in introductory calculus courses. The purpose of the paper is to investigate many of the basic manipulative properties of the integral.

The Riemann integral has many flaws, some that becomes visible in the fundamental theorem of calculus. The main point of this essay is to introduce the gauge integral, and prove a much more suitable version of that theorem. / Riemannintegralen har många brister. Vissa utav dessa ser man i integralkalkylens huvudsats. Huvudmålet med denna uppsats är att introducera gauge integralen och visa en mer lämplig version av huvudsatsen.

Thesis (M.S. in Applied Mathematics)--Naval Postgraduate School, September 2002. / Thesis advisor(s): Chris Frenzen, Bard Mansager. Includes bibliographical references (p. 49). Also available online.

Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 2006 / Includes bibliographical references (p. 214-215).

A presente pesquisa tem por objetivos identificar as contribuições: 1) da Teoria de Registros de Representação Semiótica, aliada à Teoria das Situações Didáticas, para a conceitualização da Integral de Riemann para funções de uma variável real (chamada de Integral Definida, ao longo do texto); 2) da utilização dos Mapas Conceituais para acompanhar o desenvolvimento da conceitualização da Integral Definida. É uma pesquisa qualitativa, de natureza exploratória, que tem a Engenharia Didática como metodologia de pesquisa. Os pressupostos teóricos estão alicerçados às Teorias das Situações Didáticas e à Teoria de Registro de Representação Semiótica. Os Mapas Conceituais foram usados como instrumento didático de acompanhamento. Após pesquisa bibliográfica, foi elaborada e aplicada uma sequência didática por meio de minicurso, em um grupo de treze alunos de 1º a 3º períodos dos cursos de Engenharia Civil, de Engenharia Ambiental, Engenharia de Produção e Licenciatura em Química. Na sequência didática, aplicada no período de 06 a 15 de agosto de 2012, foi solicitada, em cinco momentos, a elaboração e/ou alteração de um Mapa Conceitual, no software Cmap Tools, envolvendo os conteúdos das atividades. Os softwares Geogebra e wxMaxima foram usados como recursos didáticos na resolução das atividades propostas. Foi observado que a discussão em grupo e a exploração computacional trazem importantes contribuições para a aprendizagem dos conceitos. Além disso, pode-se inferir que a análise dos Mapas Conceituais elaborados pelos alunos, permite, ao professor, conhecer os aspectos que devem ser reforçados para que a compreensão do conteúdo integral definida seja amplificada, identificar possíveis obstáculos didáticos e confrontar conceitos-chave de docentes e discentes. Observou-se, ainda, que, as atividades planejadas a fim de proporcionar autonomia e ação pelo aluno, baseadas nos diferentes registros de representação semiótica, tornam-se mais propícias à atribuição de significação ao conteúdo ensinado. / The purposes of the present research are to identify the contributions of: 1) the theory of Semiotic Representation Registers, allied to theory of Didactic Situations, to the construction of the concept of Riemann‘s Integral for functions of a real variable (named as Definite Integral along the text); 2) the use of Concept Maps in order to notice the development of conceptualization of Definite Integral. It is a qualitative research, of exploratory nature, which has adopted the Didactical Engineering as research methodology. The theoretical support is based on Theories of Didactic Situations and Theory of Semiotic Representation Registers. For monitoring, the Concept Maps were used as didactic instruments. After literature review, it was developed and applied a didactic sequence, through a mini course for thirteen students of 1st to 3rd grade levels of Civil Engineering, Environmental Engineering, Production Engineering and License in Chemistry. During the didactic sequence applied from 6th to 15th of August of 2012, on five occasions, it was required the production and/or alteration of a Concept Map, in the software Cmap Tools, involving the contents of the activities. The softwares Geogebra and wxMaxima were used as didactic resources for solving the respective activities. It was observed that group discussion and computational exploration can bring about important contributions for the concepts learning. Furthermore, the analysis of the Concept Maps produced by students, allows the teacher to recognize features that should be strengthened to improve the understanding of Definite Integral content. It also allows to identify possible didactic difficulties, and to confront key-concepts of teachers and students. It was noticed that, activities planned in order to provide autonomy and attitude in the students, based on different registers of semiotic representation, are more likely to provide meanings to the taught contents.

A presente pesquisa tem por objetivos identificar as contribuições: 1) da Teoria de Registros de Representação Semiótica, aliada à Teoria das Situações Didáticas, para a conceitualização da Integral de Riemann para funções de uma variável real (chamada de Integral Definida, ao longo do texto); 2) da utilização dos Mapas Conceituais para acompanhar o desenvolvimento da conceitualização da Integral Definida. É uma pesquisa qualitativa, de natureza exploratória, que tem a Engenharia Didática como metodologia de pesquisa. Os pressupostos teóricos estão alicerçados às Teorias das Situações Didáticas e à Teoria de Registro de Representação Semiótica. Os Mapas Conceituais foram usados como instrumento didático de acompanhamento. Após pesquisa bibliográfica, foi elaborada e aplicada uma sequência didática por meio de minicurso, em um grupo de treze alunos de 1º a 3º períodos dos cursos de Engenharia Civil, de Engenharia Ambiental, Engenharia de Produção e Licenciatura em Química. Na sequência didática, aplicada no período de 06 a 15 de agosto de 2012, foi solicitada, em cinco momentos, a elaboração e/ou alteração de um Mapa Conceitual, no software Cmap Tools, envolvendo os conteúdos das atividades. Os softwares Geogebra e wxMaxima foram usados como recursos didáticos na resolução das atividades propostas. Foi observado que a discussão em grupo e a exploração computacional trazem importantes contribuições para a aprendizagem dos conceitos. Além disso, pode-se inferir que a análise dos Mapas Conceituais elaborados pelos alunos, permite, ao professor, conhecer os aspectos que devem ser reforçados para que a compreensão do conteúdo integral definida seja amplificada, identificar possíveis obstáculos didáticos e confrontar conceitos-chave de docentes e discentes. Observou-se, ainda, que, as atividades planejadas a fim de proporcionar autonomia e ação pelo aluno, baseadas nos diferentes registros de representação semiótica, tornam-se mais propícias à atribuição de significação ao conteúdo ensinado. / The purposes of the present research are to identify the contributions of: 1) the theory of Semiotic Representation Registers, allied to theory of Didactic Situations, to the construction of the concept of Riemann‘s Integral for functions of a real variable (named as Definite Integral along the text); 2) the use of Concept Maps in order to notice the development of conceptualization of Definite Integral. It is a qualitative research, of exploratory nature, which has adopted the Didactical Engineering as research methodology. The theoretical support is based on Theories of Didactic Situations and Theory of Semiotic Representation Registers. For monitoring, the Concept Maps were used as didactic instruments. After literature review, it was developed and applied a didactic sequence, through a mini course for thirteen students of 1st to 3rd grade levels of Civil Engineering, Environmental Engineering, Production Engineering and License in Chemistry. During the didactic sequence applied from 6th to 15th of August of 2012, on five occasions, it was required the production and/or alteration of a Concept Map, in the software Cmap Tools, involving the contents of the activities. The softwares Geogebra and wxMaxima were used as didactic resources for solving the respective activities. It was observed that group discussion and computational exploration can bring about important contributions for the concepts learning. Furthermore, the analysis of the Concept Maps produced by students, allows the teacher to recognize features that should be strengthened to improve the understanding of Definite Integral content. It also allows to identify possible didactic difficulties, and to confront key-concepts of teachers and students. It was noticed that, activities planned in order to provide autonomy and attitude in the students, based on different registers of semiotic representation, are more likely to provide meanings to the taught contents.

The object of this paper is to define, to establish necessary and sufficient conditions for the existence of, and to consider the elementary properties of the Riemann definite integral of a bounded function.

Propaedeutics of Differential and Integral Calculi Author: Ing. Martin Malachov Department: Department of Mathematics Education Supervisor: Zdeněk Halas, DiS., Ph.D. Department of Mathematics Education Keywords: propaedeutics, derivative, Riemann integral, applications Differential and integral calculi both are interesting and beautiful branches of mathematics with many interdisciplinary overlaps and significant and practi- cal applications. Learning as well as teaching of these topics is very difficult and demanding. In this thesis we show that the derivatives and integrals have much to offer in the high school education while the schooling can be eased and made attractive with intentional propaedeutics and knowledge of rich background of applications. First part of the thesis presents short contem- plation on the current state of teaching and literature, we focus on the urge for the propaedeutics of the differential and integral calculi revealed by us. We identify key terms that can be used to build useful preconcepts during the whole high school education, even in the elementary education. In the latter part of the thesis we offer teacher innovative texts and a rich set of origi- nal exercises that can be used for motivation, application and propaedeutics of the differential and integral calculi. We also present...