Wednesday, November 9, 2011

calculus - simpson's rule

Download (GSP 5 file): calculus_SimpsonsRule.gsp

This sketch shows the area measured by Simpson's Rule using parabolic arcs. The area beneath a parabolic arc was known to the Greeks ages before calculus was used, and Simpson's Rule is a slick application of this fact. The interval is divided into an even number of sub-intervals and a parabolic arc is chosen for each sub-interval.  The application of the rule does not of course involve actually finding the parabolas, but I found this graphical exploration illuminating.  

3 comments:

  1. Nate,
    I always thought it was fascinating that approximating a Cubic by this Quadratic curve, gives an exact solution... math is just too darn kind.

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  2. Lou Talman explains how Simpson' Rule is exact for quintics as well. By "exact", he means that the error of the primary expression can be found exactly.

    clem.mscd.edu/~talmanl/PDFs/Misc/Quintics.pdf

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  3. I suppose that this would make it perfect for quartics as well, which could be regarded as degenerate quintics.

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