Hey matdavies.

In terms of approximations, typically the way that numerical schemes work is that they use more terms in the Taylor series expansion, or they fit some kind of interpolating function (like a spline).

What I would suggest you do is take the Taylor series expansion of your function that you are integrating with an appropriate center and then look at how the higher order terms are expressed and see how quickly they vanish.

If they don't vanish quick, this will tell you that you need more terms or a particular form of integration that accounts for this error and you should read the literature or available books for such techniques.