MAIN POINTS
Simply using one instance of Newton-Cotes over a large domain is impractical and full of error. Dividing the domain up into sections and doing a Newton-Cotes integration of those pieces is much more accurate. The approximation is thus a sum of however many polynomials there are from splitting up the domain. The error term is the sum of each error term. The composite midpoint rule is described by Thm 4.6 as having a pretty error term. The midpoint rule is a common rule in calculus classes. This rule is run on a sample equation, and they practice on some Maple code.
CHALLENGES
Why do they show the Composite Midpoint Rule after the Composite Simpson's rule? I'd think the Midpoint rule would be the simpler one... everything else seems to follow directly from the Newton-Cotes formulas (error terms etc).
REFLECTION
I'd like to try this is discrete data points with no explicit formula. There is then a limit to the number of divisions you can make, and there is no error bound because you can't be sure of the derivative at any given point.
This seems similar to what Excel might use to draw continuous graphs out of discrete points— but the graph in Fig 4.7 has abrupt edges in the graph (instantaneous derivative change), whereas Excel produces continuous graphs.
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