As I mentioned in my comment, the easiest way to integrate a list of values is to use `Interpolation` with `Integrate` or `NIntegrate`. As [Jens noted](http://mathematica.stackexchange.com/questions/26882/simpsons-rule-with-listconvolve-or-listcorrelate#comment83803_26882), the upgrade information for `ListIntegrate`, an obsolete function also mentions the same thing. For the sake of completeness, here's how you'd do it:

    With[{if = Interpolation[(* list of {x, y} pairs *), InterpolationOrder -> k}, 
        Integrate[if[x], {x, Sequence @@ if["Domain"][[1]]}]
    ]