calculus and analysis - Evaluating a surface integral

I'm trying to compute the integral

$$\int_S x^2+z^2\,{\rm d}S,$$ where $S$ is the surface $$S\colon~ \frac{x^2}{2} + \frac{y^2}{3} + \frac{z^2}{2} = 1, \quad y \geq 0.$$

One possible parametrization is:

$${\bf x}(u,v) = (\sqrt{2} \cos u \cos v, \sqrt{3} \cos u \sin v, \sqrt{2}\sin u),$$ with $-\frac \pi 2 \leq u \leq \frac \pi 2, 0 \leq v \leq \pi$. Then I make:

X[u_,v_] :=  {Sqrt[2] Cos[u] Cos[v], Sqrt[3] Cos[u] Sin[v], Sqrt[2] Sin[u]}


f[{x_, y_, z_}] := x^2 + z^2

Integrate[f[X[u, v]] Norm[Cross[D[X[u, v], u], D[X[u, v], v]]], {u, -π/2, π/2}, {v, 0 , π}]

but Mathematica just won't compute it (keeps running on and on). What is an efficient way to do this? Thanks.

Answer

Solution with NIntegrate

NIntegrate[f[X[u, v]] Norm[Cross[D[X[u, v], u], D[X[u, v], v]]], {u, -π/2, π/2}, {v, 0, π}]
(* 19.8097 *)

An alternative approach to the problem is

s = ImplicitRegion[{x^2/2 + y^2/3 + z^2/2 == 1 && y > 0}, {x, y, z}];
Chop[NIntegrate[x^2 + z^2, {x, y, z} ∈ s], 10^-7]

which, of course, yields the same answer.

Added: Solution with Integrate

Consider the calculation in cylindrical coordinates with the axis along y. Integrating about the axis of symmetry then leaves the integral of 2 π r^3 over the 1D region,

s1 = ImplicitRegion[{r^2/2 + y^2/3 == 1 && r > 0 && y > 0}, {r, y}];
2 π Integrate[r^3, {r, y} ∈ s1]
(* 1/2 π (10 + 3 Sqrt[2] ArcCot[Sqrt[2]]) *)

with the numerical value given earlier.