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,, where for. Here we require thatbe even. Over each interval, for, we approximate f ( x ) with a quadratic curve that interpolates the points, , and.
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. Note that the following quadratic function interpolates the three points, , and:
Since this function is unique, this must be the quadratic function with which we approximate f ( x ) on. Also, if the three interpolating points all lie on the same line, then this function reduces to a linear function. Therefore, since for each i ,
By evaluating the integral on the right, we obtain
Summing the definite integrals over each interval, for, provides the approximation
By simplifying this sum we obtain the approximation.