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.

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.

Nate,

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

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.

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

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

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