Editing Newton–Cotes formulas (section)

== Description ==
It is assumed that the value of a function {{mvar|f}} defined on <math>[a, b]</math> is known at <math>n + 1</math> equally spaced points: <math>a \leq x_0 < x_1 < \dots < x_n \leq b</math>. There are two classes of Newton–Cotes quadrature: they are called "closed" when <math>x_0 = a</math> and <math>x_n = b</math>, i.e. they use the function values at the interval endpoints, and "open" when <math>x_0 > a</math> and <math>x_n < b</math>, i.e. they do not use the function values at the endpoints. Newton–Cotes formulas using <math>n+1</math> points can be defined (for both classes) as<ref>{{cite book |author-link1=Alfio Quarteroni |last1=Quarteroni |first1=Alfio |last2=Sacco |first2=Riccardo |last3=Saleri |first3=Fausto |title=Numerical Mathematics |date=2006 |publisher=Springer |isbn=978-3-540-34658-6 |pages=386–387 |edition=Second}}</ref>
<math display="block">\int_a^b f(x)\, dx \approx \sum_{i = 0}^n w_i\, f(x_i),</math>
where
*for a closed formula, <math>x_i = a + ih</math>, with <math>h = \frac{b - a}{n}</math>,
*for an open formula, <math>x_i = a + (i + 1)h</math>, with <math>h = \frac{b - a}{n + 2}</math>.

The number {{mvar|h}} is called ''step size'', <math>w_i</math> are called ''weights''. The weights can be computed as the integral of [[Lagrange polynomial|Lagrange basis polynomial]]s. They depend only on <math>x_i</math> and not on the function {{mvar|f}}. Let <math>L(x)</math> be the interpolation polynomial in the Lagrange form for the given data points <math>(x_0, f(x_0)), (x_1, f(x_1)), \ldots, (x_n, f(x_n))</math>, then
<math display="block"> \int_a^b f(x)\, dx \approx \int_a^b L(x)\, dx = \int_a^b \left(\sum_{i = 0}^n f(x_i) l_i(x)\right)\, dx
= \sum_{i = 0}^n f(x_i) \underbrace{\int_a^b l_i(x)\, dx}_{w_i}.</math>