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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 errors.

If we use the second order Lagrange interpolation formula with a fixed slice width :

and integrate from x_i to x_i+2 the area is

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.

References & Web Resources

• Press et al, Numerical Recipes...

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