# Simpson’s 3/8 rule

Simpson’s $\frac{3}{8}$ rule is a method for approximating a definite integral by evaluating the integrand at finitely many points. The formal rule is given by

 $\int_{x_{0}}^{x_{3}}f(x)\,dx\;\approx\;\frac{3h}{8}\left[f(x_{0})+3f(x_{1})+3f% (x_{2})+f(x_{3})\right]$

where $x_{1}=x_{0}+h$, $x_{2}=x_{0}+2h$, $x_{3}=x_{0}+3h$.

Simpson’s $\frac{3}{8}$ rule is the third Newton-Cotes quadrature formula. It has degree of precision 3. This means it is exact for polynomials of degree less than or equal to three. Simpson’s $\frac{3}{8}$ rule is an improvement to the traditional Simpson’s rule. The extra function evaluation gives a slightly more accurate approximation . We can see this with an example.

Using the fundamental theorem of the calculus, one shows

 $\int_{0}^{\pi}\sin(x)\,dx=2.$

In this case Simpson’s rule gives,

 $\int_{0}^{\pi}\sin(x)\,dx\,\approx\;\frac{\pi}{6}\left[\sin(0)+4\sin\left(% \frac{\pi}{2}\right)+\sin(\pi)\right]\,=\,2.094$

However, Simpson’s $\frac{3}{8}$ rule does slightly better.

 $\int_{0}^{\pi}\sin(x)\,dx\,\approx\;\left(\frac{3}{8}\right)\frac{\pi}{3}\left% [\sin(0)+3\sin\left(\frac{\pi}{3}\right)+3\sin\left(\frac{2\pi}{3}\right)+\sin% (\pi)\right]\,=\,2.040$
Title Simpson’s 3/8 rule Simpsons38Rule 2013-03-22 13:40:56 2013-03-22 13:40:56 Daume (40) Daume (40) 11 Daume (40) Definition msc 41A05 msc 41A55