Transcription of The Riemann Integral - University of California, Davis
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The Riemann Integral - University of California, Davis
Chapter 1 The Riemann IntegralI know of some universities in England where the Lebesgue Integral istaughtin the first year of a mathematics degree instead of the Riemannintegral, but I know of no universities in England where studentslearnthe Lebesgue Integral in the first year of a mathematics degree. (Ap-proximate quotation attributed to T. W. K orner)Letf: [a, b] Rbe a bounded (not necessarily continuous) function on acompact (closed, bounded) interval. We will define what it means forfto beRiemann integrable on [a, b] and, in that case, define its Riemann Integral offon [a, b] is a real number whose geometrical interpretation is thesigned area under the graphy=f(x) fora x b.
1.1. Definition of the Riemann integral 3 If P = {I1,I2,...,In} is a partition of I, let Mk = sup Ik f, mk = inf Ik f. These suprema and infima are well-defined, finite real numbers since f is bounded.
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