Question

You have an unknown function that is monotone increasing for

1 < x < 5 and have the following information about the function values. With the clear understanding that there is no way to get an exact integral, how would you try and approximate the area under the curve?


1 3.170
1.2 3.505
1.6 4.317
2.8 8.516
2.9 9.041
3.3 11.528
3.6 13.885
4 17.873
4.2 20.312
4.8 29.982
5 32.724

Answer 1

You could use the Trapezium Rule and then use Richardson Extrapolation to extrapolate to zero step size. If the step size is very large you need to use extrapolation using rational functions, like Pade approximants to do the extrapolation.