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).