# 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

$${\int}_{{x}_{0}}^{{x}_{3}}f(x)\mathit{d}x\approx \frac{3h}{8}\left[f({x}_{0})+3f({x}_{1})+3f({x}_{2})+f({x}_{3})\right]$$ |

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

$${\int}_{0}^{\pi}\mathrm{sin}(x)\mathit{d}x=2.$$ |

In this case Simpson’s rule gives,

$${\int}_{0}^{\pi}\mathrm{sin}(x)\mathit{d}x\approx \frac{\pi}{6}\left[\mathrm{sin}(0)+4\mathrm{sin}\left(\frac{\pi}{2}\right)+\mathrm{sin}(\pi )\right]=\mathrm{\hspace{0.17em}2.094}$$ |

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

$${\int}_{0}^{\pi}\mathrm{sin}(x)\mathit{d}x\approx \left(\frac{3}{8}\right)\frac{\pi}{3}\left[\mathrm{sin}(0)+3\mathrm{sin}\left(\frac{\pi}{3}\right)+3\mathrm{sin}\left(\frac{2\pi}{3}\right)+\mathrm{sin}(\pi )\right]=\mathrm{\hspace{0.17em}2.040}$$ |

Title | Simpson’s 3/8 rule |
---|---|

Canonical name | Simpsons38Rule |

Date of creation | 2013-03-22 13:40:56 |

Last modified on | 2013-03-22 13:40:56 |

Owner | Daume (40) |

Last modified by | Daume (40) |

Numerical id | 11 |

Author | Daume (40) |

Entry type | Definition |

Classification | msc 41A05 |

Classification | msc 41A55 |