Integrales indefinidas y definidas Antiderivadas de funciones reales de variable real. la integral indefinida. Sumas de Riemann. la integral definida. Teorema fundamental del cálculo. Métodos de integración. Aplicaciones de las integrales al cálculo de áreas y volúmenes. Aplicaciones en la física y otras disciplinas
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2022-10-19
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Universidad Nacional de Educación Enrique Guzmán y Valle
Abstract
El objetivo de este trabajo de investigación es ayudar a su comprensión, antes de desarrollar el tema en sí, en el primer capítulo presentamos los prerrequisitos: funciones, representación de modelos y
una breve referencia histórica: los aportes de los matemáticos griegos de los siglos IV y II
a. C., como Eudoxo y Arquímedes, así como Cavalieri en el siglo VXII, que sentaron
procedimientos que usaron los que formularon el cálculo infinitesimal al final del siglo
XVII: Newton y Leibniz. Sus teorías fueron cuestionadas por el obispo Berkeley en su
famoso libro El Analista, en el cual muestra fundamentos sólidos de inconsistencias en la
base de esta teoría. Se dice que el primero que utiliza 𝜀����� − 𝛿����� en sus demostraciones es
Augustin Louis Cauchy y el primero en dar una definición formal es Bernard Bolzano. Sin
embargo, es Karl Weierstrass el que formula con rigor científico una definición de límite
de una función en un punto de la variable independiente, corrigiendo las observaciones de
los detractores del cálculo infinitesimal. Después de esta reformulación, la teoría se
denominó cálculo diferencial e integral, para referirse a las derivadas y a las integrales. El Capítulo II aborda el núcleo de este estudio: la integral definida, que la
definimos como el área determinada por la función, el eje x en un intervalo acotado y
cerrado [a, b]. Las antiderivadas de funciones reales de variable real, las sumas de
Riemann, el teorema fundamental del cálculo y aplicaciones de las integrales a la Física,
así como al cálculo de áreas y volúmenes.
En el Capítulo III realizamos una síntesis de nuestro estudio y, finalmente, en el
Capítulo IV formulamos una aplicación didáctica para la educación secundaria. Según las disposiciones sobre el contenido monográfico, incluimos la aplicación
didáctica del tema en mención en el nivel de educación superior, principalmente en la
formación docente, así como la síntesis del trabajo, una apreciación crítica, y las
respectivas conclusiones y sugerencias que se desprenden del presente estudio.
The objective of this research work is to help your understanding, before developing the topic itself, in the first chapter we present the prerequisites: functions, representation of models and a brief historical reference: the contributions of the Greek mathematicians of the 4th centuries and II a. C., like Eudoxo and Archimedes, as well as Cavalieri in the 52nd century, who established procedures used by those who formulated infinitesimal calculus at the end of the 17th century: Newton and Leibniz. His theories were questioned by Bishop Berkeley in his famous book The Analyst, in which he shows solid foundations of inconsistencies at the basis of this theory. It is said that the first to use 𝜀���� − 𝛿���� in his proofs is Augustin Louis Cauchy and the first to give a formal definition is Bernard Bolzano. However, it is Karl Weierstrass who formulates with scientific rigor a definition of the limit of a function at a point of the independent variable, correcting the observations of the detractors of infinitesimal calculus. After this reformulation, the theory was called differential and integral calculus, to refer to the derivatives and integrals. Chapter II addresses the core of this study: the definite integral, which we define as the area determined by the function, the x axis in a bounded and closed interval [a, b]. The antiderivatives of real functions of a real variable, the Riemann sums, the fundamental theorem of calculus and applications of integrals to Physics, as well as the calculus of areas and volumes. In Chapter III we make a synthesis of our study and, finally, in Chapter IV we formulate a didactic application for secondary education. According to the provisions on the monographic content, we include the didactic application of the topic in question at the level of higher education, mainly in teacher training, as well as the synthesis of the work, a critical appreciation, and the respective conclusions and suggestions that arise from the present study.
The objective of this research work is to help your understanding, before developing the topic itself, in the first chapter we present the prerequisites: functions, representation of models and a brief historical reference: the contributions of the Greek mathematicians of the 4th centuries and II a. C., like Eudoxo and Archimedes, as well as Cavalieri in the 52nd century, who established procedures used by those who formulated infinitesimal calculus at the end of the 17th century: Newton and Leibniz. His theories were questioned by Bishop Berkeley in his famous book The Analyst, in which he shows solid foundations of inconsistencies at the basis of this theory. It is said that the first to use 𝜀���� − 𝛿���� in his proofs is Augustin Louis Cauchy and the first to give a formal definition is Bernard Bolzano. However, it is Karl Weierstrass who formulates with scientific rigor a definition of the limit of a function at a point of the independent variable, correcting the observations of the detractors of infinitesimal calculus. After this reformulation, the theory was called differential and integral calculus, to refer to the derivatives and integrals. Chapter II addresses the core of this study: the definite integral, which we define as the area determined by the function, the x axis in a bounded and closed interval [a, b]. The antiderivatives of real functions of a real variable, the Riemann sums, the fundamental theorem of calculus and applications of integrals to Physics, as well as the calculus of areas and volumes. In Chapter III we make a synthesis of our study and, finally, in Chapter IV we formulate a didactic application for secondary education. According to the provisions on the monographic content, we include the didactic application of the topic in question at the level of higher education, mainly in teacher training, as well as the synthesis of the work, a critical appreciation, and the respective conclusions and suggestions that arise from the present study.
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Amarillo Rimari, E. R. (2022). Integrales indefinidas y definidas
Antiderivadas de funciones reales de variable real. la integral indefinida.
Sumas de Riemann. la integral definida. Teorema fundamental del
cálculo. Métodos de integración. Aplicaciones de las integrales al cálculo
de áreas y volúmenes. Aplicaciones en la física y otras disciplinas (monografia de pregrado). Universidad Nacional de Educación Enrique Guzmán y Valle, Lima, Perú