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Simpsonin sääntö antaa tarkan vastauksen kaikille polynomeille, joiden asteluku on pienempi tai yhtä suuri kuin kolme, koska virhetermi sisältää funktion neljännen derivaatan, mikä on tietysti = 0 kun funktio f on korkeintaan kolmatta astetta.
Since the error term is proportional to the fourth derivative of f {\displaystyle f} at ξ {\displaystyle \xi } , this shows that Simpson's rule provides exact results for any polynomial f {\displaystyle f} of degree three or less, since the fourth derivative of such a polynomial is zero at all points.
Se approksimoi integraalia ∫ x 1 x 5 f ( x ) d x {\displaystyle \int _{x_{1}}^{x_{5}}f(x)\,dx} käyttämällä arvoja viidessä tasaisesti jaetussa pisteessä. x 1 , x 2 = x 1 + h , x 3 = x 1 + 2 h , x 4 = x 1 + 3 h , x 5 = x 1 + 4 h . {\displaystyle x_{1},\quad x_{2}=x_{1}+h,\quad x_{3}=x_{1}+2h,\quad x_{4}=x_{1}+3h,\quad x_{5}=x_{1}+4h.\,} Abramowitzin ja Stegunin kirjassa Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables on mainittu, että ∫ x 1 x 5 f ( x ) d x = 2 h 45 ( 7 f ( x 1 ) + 32 f ( x 2 ) + 12 f ( x 3 ) + 32 f ( x 4 ) + 7 f ( x 5 ) ) + virhetermi , {\displaystyle \int _{x_{1}}^{x_{5}}f(x)\,dx={\frac {2h}{45}}\left(7f(x_{1})+32f(x_{2})+12f(x_{3})+32f(x_{4})+7f(x_{5})\right)+{\text{virhetermi}},} ja virhetermi on − 8 945 h 7 f ( 6 ) ( c ) {\displaystyle -\,{\frac {8}{945}}h^{7}f^{(6)}(c)} jollakin luvulla c väliltä x 1 ≤ c ≤ x 5 {\displaystyle x_{1}\leq c\leq x_{5}} .
It approximates an integral ∫ x 1 x 5 f ( x ) d x {\displaystyle \int _{x_{1}}^{x_{5}}f(x)\,dx} by using the values of ƒ at five equally spaced points x 1 , x 2 = x 1 + h , x 3 = x 1 + 2 h , x 4 = x 1 + 3 h , x 5 = x 1 + 4 h . {\displaystyle x_{1},\quad x_{2}=x_{1}+h,\quad x_{3}=x_{1}+2h,\quad x_{4}=x_{1}+3h,\quad x_{5}=x_{1}+4h.\,} It is expressed thus in Abramowitz and Stegun (1972, pp. 886): ∫ x 1 x 5 f ( x ) d x = 2 h 45 ( 7 f ( x 1 ) + 32 f ( x 2 ) + 12 f ( x 3 ) + 32 f ( x 4 ) + 7 f ( x 5 ) ) + error term , {\displaystyle \int _{x_{1}}^{x_{5}}f(x)\,dx={\frac {2h}{45}}\left(7f(x_{1})+32f(x_{2})+12f(x_{3})+32f(x_{4})+7f(x_{5})\right)+{\text{error term}},} and the error term is − 8 945 h 7 f ( 6 ) ( c ) {\displaystyle -\,{\frac {8}{945}}h^{7}f^{(6)}(c)} for some number c between x1 and x5.
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