##### Actions

$$\int\limits _ { a } ^ { b } f ( x) dx \cong \ ( b - a) \sum _ {k = 0 } ^ { n } B _ {k} ^ {(} n) f ( x _ {k} ^ {(} n) )$$

for the computation of an integral over a finite interval $[ a, b]$, with nodes $x _ {k} ^ {(} n) = a + kh$, $k = 0 \dots n$, where $n$ is a natural number, $h = ( b - a)/n$, and the number of nodes is $N = n + 1$. The coefficients are determined by the fact that the quadrature formula is interpolational, that is,

$$B _ {k} ^ {(} n) = \ \frac{(- 1) ^ {n - k } }{k! ( n - k)! n } \int\limits _ { 0 } ^ { n } \frac{t ( t - 1) \dots ( t - n) }{t - k } dt.$$

For $n = 1 \dots 7 , 9$ all coefficients are positive, for $n = 8$ and $n \geq 10$ there are both positive and negative ones among them. The algebraic degree of accuracy (the number $d$ such that the formula is exact for all polynomials of degree at most $d$ and not exact for $x ^ {d + 1 }$) is $n$ for odd $n$ and $n + 1$ for even $n$. The simplest special cases of the Newton–Cotes quadrature formula are: $n = 1$, $h = b - a$, $N = 2$,

$$\int\limits _ { a } ^ { b } f ( x) dx \cong \ { \frac{b - a }{2} } [ f ( a) + f ( b)],$$

the trapezium formula; $n = 2$, $h = ( b - a)/2$, $N = 3$,

$$\int\limits _ { a } ^ { b } f ( x) dx \cong \ { \frac{b - a }{6} } \left [ f ( a) + 4f \left ( { \frac{a + b }{2} } \right ) + f ( b) \right ] ,$$

the Simpson formula; $n = 3$, $h = ( b - a)/3$, $N = 4$,

$$\int\limits _ { a } ^ { b } f ( x) dx \cong \ { \frac{b - a }{8} } \left [ f ( a) + 3f ( a + h) + 3f ( a + 2h) + f ( b) \right ] ,$$

the "three-eighths" quadrature formula. For large $n$ the Newton–Cotes formula is seldom used (because of the property of the coefficients for $n \geq 10$ mentioned above). One prefers to use for small $n$ the compound Newton–Cotes quadrature formulas, namely, the trapezium formula and Simpson's formula.

The coefficients of the Newton–Cotes quadrature formula for $n$ from 1 to 20 are listed in [3].

The formula first appeared in a letter from I. Newton to G. Leibniz in 1676 (see [1]) and later in the book [2] by R. Cotes, where the coefficients of the formula are given for $n$ from 1 to 10.

#### References

 [1] I. Newton, "Mathematical principles of natural philosophy" A.N. Krylov (ed.) , Collected works , 7 , Moscow-Leningrad (1936) (In Russian; translated from Latin) [2] R. Cotes, "Harmonia Mensurarum" , 1–2 , London (1722) (Published by R. Smith after Cotes' death) [3] V.I. Krylov, L.T. Shul'gina, "Handbook on numerical integration" , Moscow (1966) (In Russian)