The last chapter (Chapter 11) of my second-year module G12MAN Mathematical Analysis consists of a brief (one-lecture) introduction to Riemann integration. This material is motivated in terms of questions of antidifferentiation and area. The proofs of the lemmas and theorems are not included here (see books for details), but the main definitions are given in full, along with illustrative examples and diagrams, and the statements of the main theorems. Material discussed includes: partitions of intervals; Riemann lower and upper sums (approximation using rectangles); the Riemann lower and upper integrals; Riemann integrability of functions, and the Riemann integral. Examples are given of functions which are/are not Riemann integrable: in particular, continuous real-valued functions on closed intervals are Riemann integrable. The lecture concludes with the statements of the (first) Fundamental Theorem of Calculus and the Mean Value Theorem of Integral Calculus.
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