The trapezium rule gives an estimate for the value of an integralas
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 ofSuppose that we use the trapezium rule twice with different values of
obtaining
(1) and
(2)
(1)-(2) gives
Substituting back into (2) (assumingso that
is a better estimate for I) gives
For a very simple example, consider
Takeand
so that
In factexactly, so in this case the estimate is exactly right.