Let's make up some data close to yours:

data = Table[{x, 700 x Exp[-x] + 50 RandomReal[]}, {x, 0, 9, .1}];

When you have just list of points you have three choices for integration:

- integrate data
- integrate interpolation
- integrate model fit

Let's first get all three on the same figure.

(* interpolation *)

f = Interpolation[data];

(* model fit *)

model = a x Exp[-b x];

fit = FindFit[data, model, {a, b}, x]

modelf = Function[{t}, Evaluate[model /. fit]];

Out[] = {a -> 678.552, b -> 0.892996}

Show[

Plot[{f[x], modelf[x]}, {x, 0, 9}],

ListPlot[data, PlotStyle -> Red]

, PlotRange -> All]

Now we can compute the integrals - which all are pretty close to each other. You should pick a method more appropriate to your case. The 1st one - integration of data - is the simplest. You just have to sum all the Y-values and multiply them by the step between the points - in our case it is 0.1 .

(* integrate data *)

.1 Total[data[[All, 2]]]

Out[]= 1911.23

(* integrate interpolation *)

NIntegrate[f[x], {x, 0, 9}]

Out[]= 1901.01

(* integrate model fit *)

NIntegrate[modelf[x], {x, 0, 9}]

Out[]= 1756.51