Plot[t/Sqrt[2 + t^3], {t, -1, 3}, Filling -> Axis]

ir0 = ImplicitRegion[0 > y > x/Sqrt[2 + x^3] && -1 < x < 0, {x, y}];
irf[u_] :=
 ImplicitRegion[0 < y < x/Sqrt[2 + x^3] && 0 < x < u, {x, y}]
func[v_] := N[RegionMeasure[irf[v]] - RegionMeasure[ir0]]

Plot[func[v], {v, 1, 3}]