The trapezium rule gives an estimate for the value of an integralaswith accuracy proportional toso that ifand the true value of the integral isthen
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
Substituting back into (2) (assumingso thatis a better estimate for I) gives
For a very simple example, consider
In factexactly, so in this case the estimate is very accurate.