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 ?