On 3/26/09, Eugene Salamin <gene_salamin@yahoo.com> wrote:
If I'm integrating over a circle, so that there is no boundary, then one point is as good as another, and I would expect the best approximation to be to give equal weights to each point. Now if instead, I integrate over an interval, there are some boundary effects, but deep within the interval , why would I want to do otherwise than to weight the points equally?
-- Gene
Two reasons I might (tentatively) suggest: (i) You might want to be able to predict the error a priori, subject to some assumptions about the integrand (e.g. analyticity); (ii) You might want to ensure that some smaller class be integrated exactly, or at least to within working precision (e.g. quadratic polynomials). I can't say that I've ever been very convinced by these arguments either --- if left to my own devices and in a hurry, I usually fall back on Romberg's method (which at a pinch can also be tweaked to serve for ordinary differential equations). Fred Lunnon