Proof of Simpson’s Rule

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Simpson’s Rule is used to numerically estimate the value of integrals that either cannot be or are difficult to evaluate analytically. The rule approximates a function with a collection of arcs from quadratic functions and integrate across each of these.

Proof: Let P be a partition of [a,b] into n subintervals of equal width, tex2html-wrap-inline2119, where tex2html-wrap-inline2121 for tex2html-wrap-inline2123. Here we require thatbe even. Over each interval tex2html-wrap-inline2419, for tex2html-wrap-inline2421, we approximate f(x) with a quadratic curve that interpolates the points tex2html-wrap-inline2425, tex2html-wrap-inline2137, and tex2html-wrap-inline2139.


Figure 4:   Approximating the graph of y = f(x) with parabolic arcs across successive pairs of intervals to obtain Simpson’s Rule.

Since only one quadratic function can interpolate any three (non-colinear) points, we see that the approximating function must be unique for each interval tex2html-wrap-inline2419. Note that the following quadratic function interpolates the three points tex2html-wrap-inline2425, tex2html-wrap-inline2137, and tex2html-wrap-inline2139:

eqnarray298

Since this function is unique, this must be the quadratic function with which we approximate f(x) on tex2html-wrap-inline2419. Also, if the three interpolating points all lie on the same line, then this function reduces to a linear function. Therefore, since tex2html-wrap-inline2155 for each i,

eqnarray316

By evaluating the integral on the right, we obtain

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Summing the definite integrals over each interval tex2html-wrap-inline2419, for tex2html-wrap-inline2421, provides the approximation

eqnarray337

By simplifying this sum we obtain the approximation scheme

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