In class we have learned about the Trapezoid Rule for approximating integrals that cannot be solved abstractly. As we have seen, the idea is to use the endpoints [a,b] of an interval to form a trapezoid under the function. The area of such a trapezoid can often give a good approximation of the total area under the curve. The approximation can be improved by applying the trapezoid rule to subintervals and summing the areas.

Unfortunately, the trapezoid rule does not always give a precise estimation, and if you’re estimating integrals while calculating, say, a missile trajectory, precision is imperative. Simpson’s Rule to the rescue! Developed by mathematician Thomas Simpson, Simpson’s Rule approximates integrals by interpolating a parabolic curve under functions. Applying good ol’ Lagrange interpolation on the endpoints and midpoint of an integral gives a smooth parabolic curve, which leads to the approximation:

Again, Simpson’s becomes a much better approximation if we subdivide the interval, finding parabolic curves within each interval. It’s easy to find the area under a parabola, which is why Simpson’s Rule gives us a fast and accurate estimation, key when you are relying on precise measurements and calculations. Check out some of the references for a more comprehensive explanation of Simpson’s Rule.

References:
http://en.wikipedia.org/wiki/Trapezoid_rule
http://en.wikipedia.org/wiki/Simpson%27s_Rule
http://www.mactech.com/articles/mactech/Vol.09/09.10/SimpsonsRule/index.html
http://www.libraryofmath.com/numerical-integration-with-the-simpson-rule.html