Let's say I have a chi squared distribution

Plot[Evaluate@PDF[ChiSquareDistribution[4], x], {x, 0, 10}, Filling -> Axis, Exclusions -> None]

and what I want to do is sum the f(x) values if f(x) is chi squared distribution, for {x,0,10}.

Let's take an easier example so I can really explain what I want: If my function was a simple Sin[x]

Plot[Sin[x], {x, 0, 5}, Filling -> Axis]

What I want is something like

Sum[Sin[x],{x,0,10}]

but the problem with Sum[] is that he takes x as discrete values - which I don't want. I would like it to be continuous. And I think I should repeat, this is not ArcLength it's a sum of the function values.

Thank you for any hints. If there is no other option I will just set a really small steps in the Sum. Not a really elegant solution, but it should work.