Question

How does Simpson's Rule work?

Answer 1

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}`.

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I hope that this was helpful.