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, 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]]}]
]