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.

31 Reunión Latinoamericana de Matemática Educativa (Relme), evento desarrollado en la Universidad de Medellín, Colombia, del 31 Julio al 04 de Agosto de 2017. / Esta investigación presenta el análisis de cómo se generó la génesis instrumental de la noción de integral definida en estudiantes universitarios desde el Enfoque instrumental, mediada por el Geogebra. Al contrastar los resultados esperados y obtenidos por medio de la Ingeniería didáctica, se observó que los estudiantes generaron esquemas de utilización de la integral definida al emplearla en la resolución de problemas matemáticos, evidenciando la instrumentalización e instrumentación de dicha noción matemática. / This research shows the analysis of how the instrumental genesis of definite-integral notion was generated by university students from the GeoGebra-based Instrumental Approach. When contrasting the expected results to the ones obtained by using Didactic Engineering, we could observe that the students generated schemes to use the definite integral by using it in solving mathematical problems, showing the instruments used and the instrumentation of this mathematical notion.

nÃo hà / Este trabalho tem como objetivo principal inserir, no Ensino MÃdio, a ideia intuitiva da Integral Definida, o cÃlculo de Ãreas abaixo do grÃfico de funÃÃes positivas, limitadas pelo eixo das abscissas e por retas verticais, ou atà mesmo entre funÃÃes positivas em um intervalo determinado pelo domÃnio das mesmas, por exemplo. Para facilitar o entendimento dessas ideias, aliado ao estudo de funÃÃes, pode-se fazer o uso de um recurso computacional como o Geogebra, software utilizado como ferramenta de apoio a aprendizagem nas atividades sugeridas nesse trabalho. As atividades aqui propostas destinam-se a alunos do ensino mÃdio. Destaca-se a importÃncia da introduÃÃo dessas ideias por estimular a construÃÃo de conhecimentos mais sÃlidos sobre o comportamento de funÃÃes. Acredita-se que, assim, a longo prazo, os alunos que ingressarem no Ensino Superior nas disciplinas de CÃlculo terÃo condiÃÃes melhores de compreender os conceitos necessÃrios e, assim, os Ãndices de reprovaÃÃo nessas disciplinas serÃo reduzidos. O trabalho a seguir apresenta uma proposta de atividades sobre o ensino desses tÃpicos com auxÃlio do software GeoGebra. Verificou-se que à possÃvel abrir os horizontes no Ãmbito do ensino e aprendizagem de MatemÃtica no Ensino MÃdio, com as ideias intuitivas de CÃlculo, fazendo o uso de ferramentas diversas, como a utilizaÃÃo de tecnologias apropriadas e que à possÃvel, inclusive, proporcionar aos estudantes novas tÃcnicas de ensino que favoreÃam a aprendizagem desses e demais conceitos matemÃticos. / This work aims to incorporate in high school, the intuitive idea of the Definite Integral, calculation of areas below the positive functions, bounded by the x-axis and vertical lines, or even between positive graph functions in a certain range the area of the same, for example. To facilitate the understanding of these ideas, combined with the study of functions, you can make use of a computational resource such as Geogebra, software used to support learning activities suggested in this work tool. The activities proposed here are intended for high school students. Highlights the importance of introducing these ideas to stimulate the construction of more accurate information on the behavior of functions. It is believed that thus the term, students who enter higher education in the disciplines of calculation will be better able to understand the necessary and thus the failure rates in these disciplines will be reduced concepts. The following work presents a proposal of activities for teaching these topics with the help of GeoGebra software. It was found that you can open the horizons within the teaching and learning of Mathematics in Secondary Education, with intuitive ideas of Calculus, making the use of various tools such as the use of appropriate technologies and that is even possible to provide students new teaching techniques that encourage learning these and other mathematical concepts.

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The main goal of my diploma work on the topic "The Riemann integral and its applications" is to create an overview that describes the individual use of the Riemann integral. The first three chapters deal with the history, the definition of the Riemann integral and with the calculation of integration. There is the overview of the individual use in the fourth chapter (area plane region, length plane curve, volume solid of revolution and area lateral surface solid of revolution) that is accompanied by solved examples, the examples are accompanied by graphical representation for a better idea. In the fifth chapter there is a collection of unsolved examples, for practising of the individual use. The collection is complemented by results for the reader to check the accuracy of calculation.