Riemann integral
Let be an interval of and let be a bounded function. For any finite set of points such that , there is a corresponding partition of .
Let be the set of all partitions of with . Then let be the infimum of the set of upper Riemann sums with each partition in , and let be the supremum of the set of lower Riemann sums with each partition in . If , then , so is decreasing (http://planetmath.org/IncreasingdecreasingmonotoneFunction) and is increasing (http://planetmath.org/IncreasingdecreasingmonotoneFunction). Moreover, and are bounded by . Therefore, the limits and exist and are finite. If , then is Riemann-integrable over , and the Riemann integral of over is defined by
Title | Riemann integral |
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Canonical name | RiemannIntegral |
Date of creation | 2013-03-22 11:49:24 |
Last modified on | 2013-03-22 11:49:24 |
Owner | bbukh (348) |
Last modified by | bbukh (348) |
Numerical id | 14 |
Author | bbukh (348) |
Entry type | Definition |
Classification | msc 28-00 |
Classification | msc 26A42 |
Related topic | RiemannSum |
Related topic | Integral2 |
Defines | Riemann integrable |