How does Simpson's Rule work?

Question

Ezra Adams · Accepted Answer

Let us approximate the definite integral

#int_a^b f(x)dx#

by Simpson's Rule #S_n#.

First, split the interval #[a,b]# into #n# (even) equal subintervals:

#[x_0,x_1], [x_1,x_2],[x_2,x_3],...,[x_{n-1},x_{n}]#,

where #a=x_0 < x_1 < x_2< cdots < x_n=b#.

Simpson's Rule #S_n# can be found by

#S_n=[f(x_0)+4f(x_1)+2f(x_2)+4f(x_2)+cdots+2f(x_{n-2})+4f(x_{n-1})+f(x_n)]{b-a}/{3n}#.

I hope that this was helpful.

Samantha Adkison · Answer

undefined Simpson's Rule is a numerical method used for approximating the definite integral of a function. It works by dividing the interval of integration into an even number of subintervals and approximating the area under the curve within each subinterval using quadratic polynomials. These quadratic polynomials are obtained by interpolating the function at three points: the endpoints of each subinterval and their midpoint. Simpson's Rule formula involves calculating the weighted sum of these quadratic approximations over all the subintervals. The weights are determined by a specific pattern depending on the number of subintervals used. This method provides a more accurate approximation of the integral compared to simpler methods like the Trapezoidal Rule, especially for functions that are well-behaved and smooth over the interval of integration.