u[t_]:=(Sqrt[2.])*230*Sin[314*t]

u[1]

plot1=Plot[u[t],{t,0,0.01}]

times=Range[0,0.01,0.0001];

Length[times]

uValues=u[times];
uValues=Map[u,times];
uValues=u/@times;

Length[uValues]

{times,uValues}//MatrixForm;
{times,uValues};

timesuValues=Transpose[{times,uValues}];
timesuValues=Table[{times[[ii]],uValues[[ii]]},{ii,Length[times]}];

plot2=ListPlot[timesuValues,PlotStyle->Red]

Show[plot1,plot2]

uNoiseValues=Table[RandomReal[{-10,10}],{Length[times]}];

uNoisyValues=uValues+uNoiseValues;

plot3=ListPlot[Transpose[{times,uNoisyValues}],PlotStyle->Green]

dt=times[[2]]-times[[1]];

Sum[uNoisyValues[[ii]],{ii,Length[uNoisyValues]}]

uNoisySum=Total[uNoisyValues]

myIntegral1=uNoisySum*dt

correctIntegral=Integrate[u[t],{t,0,0.01}]

(myIntegral1-correctIntegral)/correctIntegral*100