Trapezoids provide an improved match to the shape
of curving function as compared to rectangles but they are still
are linear. A polynomial curve for the top of the slice should provide
a better match to many functions and allow for fewer slices over
which to sum, thereby speeding computation and reducing roundoff
If we use the second order Lagrange interpolation
formula with a fixed slice width :
and integrate from xi
to xi+2 the
which is the area beneath a parabola.
Summing these over all the slices leads to
This is called Simpson's rule. Note that it
requires an even number of slices.
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