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Using the Trapezium Rule Twice to Obtain an Improved Estimate for the Value of an Integral

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

(1) and(2)

(1)-(2) gives

Substituting back into (2) (assumingso thatis a better estimate for I) gives

For a very simple example, consider

Takeandso that

In factexactly, so in this case the estimate is very accurate.