Lecture

Integral Calculus: Fundamentals and Applications

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In course

MATH-101(c): Analysis I

Étudier les concepts fondamentaux d'analyse et le calcul différentiel et intégral des fonctions réelles d'une variable. ...

Description

This lecture covers the fundamentals of integral calculus, including the concept of indefinite integrals, elementary primitives, and the fundamental theorem of analysis. It also explores the application of integrable functions, continuous functions, and the mean value theorem. The instructor explains the process of finding antiderivatives, Riemann sums, and Darboux sums, emphasizing the importance of integrability criteria and limit developments. Various examples and exercises are provided to illustrate the concepts discussed.

Instructor

Sacha Friedli

Diploma of Physics, Université de Lausanne, 1999 Phd, Institut de Physique Théorique, EPFL, 1999-2004 Post-doc, IMPA (Instituto Nacional de Matemática Pura e Aplicada), Rio de Janeiro, Brasil, 2004-2006 Adjunct professor, Departamento de Matemática (Universidade Federal de Minas Gerais), 2006-2016 Collaborateur Scientifique, Cours de Mathématiques Spéciales, EPFL, Lausanne, 2016-

Official source

Ontological neighbourhood

Mathematics

Analysis: Calculus

Related concepts (32)

Antiderivative

In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function f is a differentiable function F whose derivative is equal to the original function f. This can be stated symbolically as F' = f. The process of solving for antiderivatives is called antidifferentiation (or indefinite integration), and its opposite operation is called differentiation, which is the process of finding a derivative. Antiderivatives are often denoted by capital Roman letters such as F and G.

Fundamental theorem of calculus

The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, or the cumulative effect of small contributions). The two operations are inverses of each other apart from a constant value which depends on where one starts to compute area.

Integral

In mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus, the other being differentiation. Integration started as a method to solve problems in mathematics and physics, such as finding the area under a curve, or determining displacement from velocity. Today integration is used in a wide variety of scientific fields.

Lebesgue integration

In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the X-axis. The Lebesgue integral, named after French mathematician Henri Lebesgue, extends the integral to a larger class of functions. It also extends the domains on which these functions can be defined.

Nonelementary integral

In mathematics, a nonelementary antiderivative of a given elementary function is an antiderivative (or indefinite integral) that is, itself, not an elementary function (i.e. a function constructed from a finite number of quotients of constant, algebraic, exponential, trigonometric, and logarithmic functions using field operations). A theorem by Liouville in 1835 provided the first proof that nonelementary antiderivatives exist. This theorem also provides a basis for the Risch algorithm for determining (with difficulty) which elementary functions have elementary antiderivatives.

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