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.

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. 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.