# InfoCoBuild

## Real Analysis

Real Analysis. Instructor: Prof. S.H. Kulkarni, Department of Mathematics, IIT Madras. This course discusses the fundamental concepts in real analysis. Real number system and its order completeness, sequences and series of real numbers. Metric spaces: basic concepts, continuous functions, completeness, contraction mapping theorem, connectedness, intermediate value theorem, compactness, Heine-Borel theorem. Differentiation, Taylor's theorem, Riemann integral, improper integrals, sequences and series of functions, uniform convergence, power series, Weierstrass approximation theorem, equicontinuity, Arzela-Ascoli theorem. (from nptel.ac.in)

 Lecture 38 - Integration

Go to the Course Home or watch other lectures:

 Review of Set Theory Lecture 01 - Introduction Lecture 02 - Functions and Relations Lecture 03 - Finite and Infinite Sets Lecture 04 - Countable Sets Lecture 05 - Uncountable Sets, Cardinal Numbers Sequences and Series of Real Numbers Lecture 06 - Real Number System Lecture 07 - Least Upper Bound (LUB) Axiom Lecture 08 - Sequences of Real Numbers Lecture 09 - Sequences of Real Numbers (cont.) Lecture 10 - Sequences of Real Numbers (cont.) Lecture 11 - Infinite Series of Real Numbers Lecture 12 - Series of Nonnegative Real Numbers Lecture 13 - Conditional Convergence Metric Spaces: Basic Concepts Lecture 14 - Metric Spaces: Definition and Examples Lecture 15 - Metric Spaces: Examples and Elementary Concepts Lecture 16 - Balls and Spheres Lecture 17 - Open Sets Lecture 18 - Closure Points, Limit Points and Isolated Points Lecture 19 - Closed Sets Completeness Lecture 20 - Sequences in Metric Spaces Lecture 21 - Completeness Lecture 22 - Baire Category Theorem Limits and Continuity Lecture 23 - Limit and Continuity of a Function Defined on a Metric Space Lecture 24 - Continuous Functions on a Metric Space Lecture 25 - Uniform Continuity Connectedness and Compactness Lecture 26 - Connectedness Lecture 27 - Connected Sets Lecture 28 - Compactness Lecture 29 - Compactness (cont.) Lecture 30 - Characterizations of Compact Sets Lecture 31 - Continuous Functions on Compact Sets Lecture 32 - Types of Discontinuity Differentiation Lecture 33 - Differentiation Lecture 34 - Mean Value Theorems Lecture 35 - Mean Value Theorems (cont.) Lecture 36 - Taylor's Theorem Lecture 37 - Differentiation of Vector Valued Functions Integration Lecture 38 - Integration Lecture 39 - Integrability Lecture 40 - Integrable Functions Lecture 41 - Integrable Functions (cont.) Lecture 42 - Integration as a Limit of Sum Lecture 43 - Integration and Differentiation Lecture 44 - Integration of Vector Valued Functions Lecture 45 - More Theorems on Integrals Sequences and Series of Functions Lecture 46 - Sequences and Series of Functions Lecture 47 - Uniform Convergence Lecture 48 - Uniform Convergence and Integration Lecture 49 - Uniform Convergence and Differentiation Lecture 50 - Construction of Everywhere Continuous, Nowhere Differentiable Function Lecture 51 - Approximation of a Continuous Function by Polynomials: Weierstrass Theorem Lecture 52 - Equicontinuous Family of Functions: Arzela-Ascoli Theorem