Home page for accesible maths 3 Chapter 3 contents 3.12 Elliptic integrals 3.14 Perimeter of a particular ellipse

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3.13 Simpson’s Rule

Theorem (Simpson’s Rule).

Let $f(x)$ be a suitably differentiable function on $[a,b]$ . Then

\int_{a}^{b}f(x)\,dx={{(b-a)}\over{6}}\Bigl\{f(a)+4f\Bigl({{a+b}\over{2}}\Bigr% )+f(b)\Bigr\}+E

where the error term is

E=-{{(b-a)^{5}f^{(4)}(c)}\over{2880}}

for some $c$ between $a$ and $b$ .

In many cases the error is very small and Simpson’s rule gives a good approximation. When the rule is applied to a quadratic, such as $p(x)=a_{2}x^{2}+a_{1}x+a_{0}$ , the error is zero.