Open Access
2011/2012 On Riemann Sums
Brian S. Thomson
Real Anal. Exchange 37(1): 221-242 (2011/2012).


If a sum of the form \[\sum_{i=1}^n f(\xi_i)(x_i-x_{i-1})\] is used without the familiar requirement that the sequence of points \(a=x_0, x_1, \dots, x_n=b\) is increasing, do we still get a useful approximation to the integral? With a suitable set of hypotheses the answer is yes. We give applications to change of variable formulas and the problem of characterizing derivatives.


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Brian S. Thomson. "On Riemann Sums." Real Anal. Exchange 37 (1) 221 - 242, 2011/2012.


Published: 2011/2012
First available in Project Euclid: 30 April 2012

MathSciNet: MR3016862

Primary: 26A24 , 26A42

Keywords: change of variable , derivative , Henstock-Kurzweil integral , Lebesgue integral , Riemann integral , Riemann sums , Stieltjes integral

Rights: Copyright © 2011 Michigan State University Press

Vol.37 • No. 1 • 2011/2012
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