• 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
>> Introduce students to Riemann Integration