runge[x_] = 1 / (1 + 25 x^2)

rungeData11 = Table[
{x, runge[x]},
{x, -1, +1, 0.2}
]

rungeApprox11 = InterpolatingPolynomial[rungeData11, x] // Simplify

rungePlot = Show[
Plot[runge[x], {x, -1, +1}],
ListPlot[rungeData11, PlotStyle -> Red],
Plot[rungeApprox11, {x, -1, +1}, PlotStyle -> Red]
]

rungeInt = Integrate[runge[x], {x, -1, +1}]

rungeIntApprox11 = Integrate[rungeApprox11, {x, -1, +1}]

(rungeIntApprox11 - rungeInt) / rungeInt