The trapezium rule gives an estimate for the value of an integral
as
with accuracy proportional to
so that if
and the true value of the integral is
then
We can obtain an improved estimate for the value of the integral using the trapezium rule twice with different values of
Suppose that we use the trapezium rule twice with different values of
obtaining
(1) and
(2)
(1)-(2) gives
Substituting back into (2) (assuming
so that
is a better estimate for I) gives
For a very simple example, consider
Take
and
so that
In fact
exactly, so in this case the estimate is very accurate.