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I am a statistician tasked with teaching an elementary calculus course. I am about to teach Riemann sums. The breakpoints for the rectangles (the partition) that make up the Riemann sum need not be equally spaced. I get it.

My question is this. Forget the limit. Suppose you are only allowed n rectangles. What would be the optimal placement of the breakpoints to obtain the best estimate of the definite integral? Would the answer differ if you were using a trapezoid rule or higher order to estimate the function between partition boundaries? (It seems that with trapezoid rule, "nearly linear" portions of the function would not need much attention from the partition).

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There are various kinds of adaptive quadrature, where the method you suggest is used. But for a powerful sort of general method, you might look into Gaussian quadrature which can get miraculous performance from a weighted average from a small number of points, chosen without knowledge of the function. –  André Nicolas May 2 '13 at 18:14
    
Nice question. I guess that an answer is not very easy to give, since there are numerical quadrature methods that try hard to do just that. Informally, we will need more breakpoints in areas where the function oscillates more, and less where the function oscillates less. "Oscillation" may be quantified in terms of a derivative: the first if we are using a first order method, such as trapezoid, the second if we are using a second order method, such as Simpson's, and so on. Quantify all of this precisely is beyond my grasp, though. –  Giuseppe Negro May 2 '13 at 18:15
    
Thanks for the suggestions. –  Placidia May 3 '13 at 17:44

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