Assuming that the ODE is of the form
then I would put:
They are successively higher order, in terms of the Taylor series expansion, estimations of the integral
that when averaged, with suitable weighting, optimise the overall order achieved of the estimation of this integral.
Well that's just off the top of my head but I think that's all you need to say in a nutshell - at least without writing an essay!
So you had the right idea with your answer but were only wrong in that you thought they were an estimate of when in actual fact they are estimates of .
Hope that was of some help!