# Riemann Integration

Partition and refinement of partition of a closed and bounded interval. Upper Darboux sum U(P, f) and lower Darboux sum L(P, f) and associated results. Upper integral and lower integral. Darboux’s theorem. Darboux’s definition of integration over a closed and bounded interval. Riemann’s definition of integrability. Equivalence with Darboux definition of integrability (statement only). Necessary and sufficient condition for Riemann integrability.

Concept of negligible set (or zero set) defined as a set covered by countable number of open intervals sum of whose lengths is arbitrary small. Examples of negligible sets : any subset of a negligible set, finite set, countable union of negligible sets. A bounded function on closed and bounded interval is Riemann integrable if and only if the set of points of discontinuity is negligible. Example of Riemann integrable functions.

Integrability of sum, scalar multiple, product, quotient, modulus of Riemann integrable functions. Properties of Riemann integrable functions arising from the above results.

Antiderivative (primitive or indefinite integral). Properties of Logarithmic function.

Fundamental theorem of Integral Calculus. First Mean Value theorem of integral calculus.

4,000.00 5,000.00

Course:Riemann Integration

Content:
Partition and refinement of partition of a closed and bounded interval. Upper Darboux sum U(P, f) and lower Darboux sum L(P, f) and associated results. Upper integral and lower integral. Darboux’s theorem. Darboux’s definition of integration over a closed and bounded interval. Riemann’s definition of integrability. Equivalence with Darboux definition of integrability (statement only). Necessary and sufficient condition for Riemann integrability.
• Concept of negligible set (or zero set) defined as a set covered by countable number of open intervals sum of whose lengths is arbitrary small. Examples of negligible sets : any subset of a negligible set, finite set, countable union of negligible sets. A bounded function on closed and bounded interval is Riemann integrable if and only if the set of points of discontinuity is negligible. Example of Riemann integrable functions.
• Integrability of sum, scalar multiple, product, quotient, modulus of Riemann integrable functions. Properties of Riemann integrable functions arising from the above results.
• Antiderivative (primitive or indefinite integral). Properties of Logarithmic function.
• Fundamental theorem of Integral Calculus. First Mean Value theorem of integral calculus.

Syllabus: B.Sc. Mathematics Hons. courses

University: All Indian Universities

Learning Objective:
>>
Introduce students to Riemann Integration

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Video Language: English 6 months Any android smart phone/Tab Live & interactive CLASS B.Sc. Mathematics General courses

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