Clear[s, P];
P[s_] = {-2 s, 3 s, 2 s^2};
unittan[s_] = P'[s]/Sqrt[P'[s] . P'[s]];
mainunitnormal[s_] = N[unittan'[s]/Sqrt[Expand[unittan'[s] . unittan'[s]]]];
Clear[binormal];
binormal[s_] = Cross[unittan[s], mainunitnormal[s]];
Clear[hornskin, r];
r[s_] = 0.5 s^2;
hornskin[s_, t_] = Together[P[s] + r[s] Cos[t] mainunitnormal[s] + r[s] Sin[t] binormal[s]];
ParametricPlot3D[Evaluate[hornskin[s, t]], {s, 0, 2}, {t, 0, 2 Pi}, AxesLabel -> {"x", "y", "z"}, ViewPoint -> CMView];


Clear[x, y, z, r, s, t];
r[s_] = 0.5 s^2;
{x[r_, s_, t_], y[r_, s_, t_], z[r_, s_, t_]} = Chop[Simplify[P[s] + r Cos[t] mainunitnormal[s] + r Sin[t] binormal[s]]];
Clear[gradx, grady, gradz, Vxyz];
gradx[r_, s_, t_] = {D[x[r, s, t], r], D[x[r, s, t], s], D[x[r, s, t], t]};
grady[r_, s_, t_] = {D[y[r, s, t], r], D[y[r, s, t], s], D[y[r, s, t], t]};
gradz[r_, s_, t_] = {D[z[r, s, t], r], D[z[r, s, t], s], D[z[r, s, t], t]};
Vxyz[r_, s_, t_] = Simplify[Simplify[TrigReduce[Det[{gradx[r, s, t], grady[r, s, t], gradz[r, s, t]}]]]];

volume = NIntegrate[Abs[Vxyz[r, s, t]], {t, 0, 2 Pi}, {s, 0, 2}, {r, 0, 0.5 s^2}]