Trapezium Rule with Error Correction

The trapezium rule estimates the area under a curve. With strip width  the estimated area will be . We can vary the number of strips and observe how the accuracy improves as strip number increases. The error between the estimates and the true value is roughly proportional to the square of the strip width . If we have two estimates  and with  and  strips respectively and the true value of the integral is  then this means that:  and Subtracting and rearranging we can estimate  as And gain a new estimate for  as The benefit of this is that  and  can be relatively small numbers of strips and still give pretty good estimates of the true value. Use the interactivity below to observe how successive Trapezium Rule estimates closely follow the quadratic error line obtained by the process above. If  and  how many evaluations of the function were required to produce ? Compare that to the number of evaluations required for . How does the value of  respond to changes in  and ?