Introduction
Simpson’s rules are numerous approximations for definite integrals in numerical analysis, named after English mathematician Thomas Simpson (1710−1761). In calculus, basically, there are two ways to approximate the value of an integral, Reimann sums and Trapezoidal sums. However, calculating the value of an integral, we need to compute the areas of a zillion rectangles or more to get a better result. Therefore, we use Simpson’s Rule, which is a way to approximate integrals without having to deal with lots of narrow rectangles.
Simpson’s 1/3 Rule
The most basic of these rules, called Simpson’s 1/3 rule, or just Simpson’s rule, reads
Introducing the step size
Because of the
Simpson’s 3/8 rule
Thomas Simpson proposed Simpson’s 3/8 rule, often known as Simpson’s second rule, as another approach for numerical integration. Rather than a quadratic interpolation, it uses a cubic interpolation. The 3/8 rule of Simpson is as follows:
Numerical Analysis
To obtain an approximation of the definite integral
The coefficients in Simpson’s Rule have the following pattern:
Example
The question is to use Simpson’s Rule with
It is easy to see that the width of each subinterval is
Substitute all these values into the Simpson’s Rule formula:
The true solution for the integral is
Hence, the error in approximating the integral is
Conclusion
Simpson’s rule is a more accurate form of numerical integration than the Trapezoidal rule, and it should always be used before trying anything more complicated.
