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Using Trapezoid and Simpson Over Extended Intervals

Here we present two formulae for the approximation of integrals over regions containing many tex2html_wrap_inline115 values. Suppose that we want to use all N+1 values of tex2html_wrap_inline115 . Over a given subinterval we will use the Trapezoid rule. The integral over the whole interval then is approximated as


so, adding the terms we find


This is equivalent to approximating the actual integrand by a series of piecewise linear segments and then integrating.

Next, suppose we utilize the Simpson's rule formula for each subinterval. The resulting approximation for the integral over the whole interval then becomes


This result is equivalent to a piecewise quadratic approximation to the function. This last composite formula is thus more accurate than the composite formula based on th Trapezoid rule.

These formulaw do not exhaust the possible polynomial approximations. One could construct formulae which are based on piecewise cubic, piecewise quartic, etc.

Comer Duncan
Tue Sep 23 17:09:49 EDT 1997