Proof of Simpson's Rule

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, wheretex2html-wrap-inline2121 fortex2html-wrap-inline2123. Here we require thatbe even. Over each intervaltex2html-wrap-inline2419, fortex2html-wrap-inline2421, we approximate f ( x ) with a quadratic curve that interpolates the pointstex2html-wrap-inline2425, tex2html-wrap-inline2137, andtex2html-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 intervaltex2html-wrap-inline2419. Note that the following quadratic function interpolates the three pointstex2html-wrap-inline2425, tex2html-wrap-inline2137, andtex2html-wrap-inline2139:


Since this function is unique, this must be the quadratic function with which we approximate f ( x ) ontex2html-wrap-inline2419. Also, if the three interpolating points all lie on the same line, then this function reduces to a linear function. Therefore, sincetex2html-wrap-inline2155 for each i ,


By evaluating the integral on the right, we obtain


Summing the definite integrals over each intervaltex2html-wrap-inline2419, fortex2html-wrap-inline2421, provides the approximation


By simplifying this sum we obtain the approximation.

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