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The animation shows the Riemann sums approximation to the integral of an unbounded function.  Naturally one can't use partitions of equal size as is the case for Riemann integrable functions.   From the web pages

of Rudolph Vyborny.

September, 1997

The Lester R. Ford Awards, established in 1964, are made annually to authors of outstanding expository papers in the MONTHLY. The awards are named for Lester R. Ford, Sr., a distinguished mathematician, editor of the MONTHLY (1942-46), and President of the Mathematical Association of America (1947-48).

Winner of the Lester R. Ford award for expository papers appearing in Volume 103 (1996) of the AMERICAN MATHEMATICAL MONTHLY:

Robert G. Bartle,  Eastern Michigan University.   Return to the Riemann Integral    pp. 625-632.

Every calculus student sees the Riemann integral, but it is not flexible or general enough for more technical problems: not enough functions are Riemann integrable, and interchanging limits with a Riemann integral is difficult in the absence of uniform convergence.   The Lebesgue integral handles these problems, but it requires the separate study of measure theory and still fails to include some important improper integrals.   In this paper, Bartle shows that a generalized Riemann integral captures the advantages of both the Riemann integral and the Lebesgue integral without incurring the major disadvantages of either of the two classical approaches.   Bartle points out that the generalized Riemann integral is more general than the Lebesgue integral, in the sense that the set of Lebesgue integrable functions is strictly contained in the set of generalized Riemann integrable functions.   The author derives a strong form of the fundamental theorem of calculus and shows how measure theory may be recovered from the theory of generalized Riemann integrals.   He demonstrates the advantages of the generalized Riemann integral in dealing with improper integrals and convergence theorems.   By the end of the paper, Bartle has presented a strong argument for replacing the Lebesgue integral with the generalized Riemann integral.

Professor Robert Bartle, who was elected a member of the London Mathematical Society on 21 November 1968, died on 18 September 2003, aged 75. He had a distinguished career at the University of Illinois and Eastern Michigan University, USA, and a long association with Mathematical Reviews (MR). He was well known to mathematics students around the world for his influential textbooks, in particular The Elements of Real Analysis and Introduction to Real Analysis (the second coauthored with Donald R. Sherbert). His researc h centred on linear operators and spectral theory and on integration theory. He remained mathematically active throughout his career and on into retirement, publishing his last book, A Modern Theory of Integration, at age 73. During his 50-year association with MR he served as a reviewer, as a member of the MR Editorial Committee (for two separate terms) and as Executive Editor from 1976-78 and again 1986-1990, when he oversaw MR with wisdom and good humour.

 July 13, 2005