This book describes the following
topics: Elementary functions and their classification, The integration of
elementary functions, The integration of rational functions, The integration of
algebraical functions and The integration of transcendental functions.
This note introduces the concepts of measures, measurable functions and
Lebesgue integrals. Topics covered includes: Measurable functions / random
variables , Dynkinís Lemma and the Uniqueness Theorem, Borel-Cantelliís First
Lemma, Independent random variables, Kolmogorovís 0-1-law, Integration of
nonnegative functions , Jordan-Hahn Decompositions, The Lebesgue-Radon-Nikodym
Theorem, The law of large numbers.
Aim of this lecture note
is to develop an understanding of the statements of the theorems and how to
apply them carefully. Major topics covered are: Measure spaces, Outer measure,
null set, measurable set, The Cantor set, Lebesgue measure on the real line,
Counting measure, Probability measures, Construction of a non-measurable set ,
Measurable function, simple function, integrable function, Reconciliation with
the integral introduced in Prelims, Simple comparison theorem, Theorems of
Fubini and Tonelli.
note covers the following topics: Integration as summation, Integration as the
reverse of differentiation, Integration using a table of anti-derivatives,
Integration by parts, Integration by substitution, Integrating algebraic
fractions, Integrating algebraic fractions, Integration using trigonometric
formulae, Finding areas by integration, Volumes of solids of revolution,
Integration leading to log functions.
book consist as a first course in the calculus. In the treatment of each topic,
the text is intended to contain a precise statement of the fundamental principle
involved, and to insure the student's clear understanding of this principle,,
without districting his attention by the discussion of a multitude of details.