f[x_, ϵ_] := 1 - ϵ/(x^2 + ϵ)

fDataGeneric7 = Table[
{x, f[x, 1/9]},
{x, -3, +4, 7/6}
];

fDataSpecificA = Table[
{x, f[x, 1/9]},
{x, -3, 0, 3/2}
];
fDataSpecificB = Table[
{x, f[x, 1/9]},
{x, 0, +4, 2}
];

Show[
Plot[
f[x, 1/9],
{x, -5, +4},
PlotRange -> All
],
ListPlot[fDataGeneric7, PlotStyle -> Black],
ListPlot[Join[fDataSpecificA, fDataSpecificB], PlotStyle -> Red]
]

fInt = Integrate[f[x, 1/4], {x, -3, +4}]
N[%]

ω3 = {1, 4, 2, 4, 2, 4, 1} / 6 / 3;
fIntNC3 = Sum[ω3[[k]] * fDataGeneric7[[k, 2]], {k, 1, 7}]   * (4 - (-3)) // N

(fIntNC3 - fInt) / fInt

ωAB = {1, 4, 1} / 6;
fIntA = Sum[ωAB[[k]] fDataSpecificA[[k, 2]], {k, 1, 3}]  * 3 // N
fIntB = Sum[ωAB[[k]] fDataSpecificB[[k, 2]], {k, 1, 3}]  * 4 // N
fIntAB = fIntA + fIntB

(fIntAB - fInt) / fInt