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The special case of the [[Newton–Cotes quadrature formula|Newton–Cotes quadrature formula]] in which two nodes are taken:

\begin{equation} \label{eq1}
\int\limits _ { a } ^ { b }  f ( x)  dx  \cong \ 
{
\frac{b - a }{2}
 } [ f ( a) + f ( b)].
\end{equation}

If the integrand  $  f $
differs strongly from a linear function, then formula \eqref{eq1} is not very exact. In this case the interval  $  [ a, b] $
is divided into  $  n $
subintervals  $  [ x _ {i} , x _ {i + 1 }  ] $, 
$  i = 0 \dots n - 1 $, 
of length  $  h = ( b- a)/n $, 
and for the calculation of the integral over  $  [ x _ {i} , x _ {i + 1 }  ] $
one uses the trapezium formula

$$ 
\int\limits _ {x _ {i} } ^ { {x _ {i  + 1} } }
f ( x)  dx  \cong \ 
{
\frac{h}{2}
 }
[ f ( x _ {i} ) + f ( x _ {i + 1 }  )].
$$

Summation of the left- and right-hand sides of this approximate equality with respect to  $  i $
from 0 to  $  n - 1 $
leads to the composite trapezium formula:

\begin{equation} \label{eq2}
\int\limits _ { a } ^ { b }  f ( x)  dx  \cong \ 
h \left [
{
\frac{f ( a) }{2}
 } +
f ( x _ {1} ) + \dots + f ( x _ {n - 1 }  ) +
{
\frac{f ( b) }{2}
 }
\right ] ,
\end{equation}

where  $  x _ {j} = a + jh $, 
$  j = 0 \dots n $. 
The quadrature formula \eqref{eq2} is also called the trapezium formula (without adding the word composite). The algebraic degree of accuracy of the quadrature formula \eqref{eq2}, as well as of \eqref{eq1}, is equal to 1. The quadrature formula \eqref{eq2} is exact for the trigonometric functions

$$ 
\cos  
\frac{2 \pi }{b - a }
 kx,\ \ 
\sin  
\frac{2 \pi }{b - a }
 kx,\ \ 
k = 0 \dots n - 1.
$$

In the case when  $  b - a = 2 \pi $, 
formula \eqref{eq2} is exact for all trigonometric polynomials of order not exceeding  $  n - 1 $; 
furthermore, its trigonometric degree of accuracy is equal to  $  n - 1 $.

If the integrand  $  f $
is twice-continuously differentiable on  $  [ a, b] $, 
then the error  $  R ( f  ) $
of the quadrature formula \eqref{eq2}, that is, the difference between the integral and the quadrature sum, is given by

$$ 
R ( f  )  = \ 
- {
\frac{b - a }{12}
 }
h  ^ {2} f ^ { \prime\prime } ( \xi ),
$$

where  $  \xi $
is a point of  $  [ a, b] $.

====Comments====
The complete formula

$$ 
\int\limits_{a}^{b} f(x)  dx = h \left [ 
\frac{f(a)}{2}
 + f( x _ {1} ) + \dots + f( x _ {n-1} )
+ \frac{f(b)}{2} \right ] -  
\frac{b-a}{12}
 h  ^ {2} f ^ { \prime\prime } ( \xi )
$$

is often referred to as the trapezoidal rule.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  F.B. Hildebrand,   "Introduction to numerical analysis" , Dover, reprint  (1974)  pp. 95ff</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W.H. Press,   B.P. Flannery,   S.A. Teukolsky,   W.T. Vetterling,   "Numerical recipes" , Cambridge Univ. Press  (1986)  pp. 105ff</TD></TR>
</table>