integration - Area for an ellipsoid

"Calculate the area for the rotation ellipsoid you get by rotating the ellipsoid $\frac{x^2}{2}+y^2 = 1$ around the x-axis."

I solved for x:

$$y = \pm \sqrt{1-\frac{x^2}{2}}$$

Then did $y = 0$ to get the integration limits, $\pm \sqrt(2)$.

So I've ended up with an integral I don't know if it's correct, and even if it is, I can't solve it.

$$4\pi \int_{\sqrt{2}}^{\sqrt{2}} \sqrt{1-\frac{x^2}{2}} \sqrt{1-\frac{x}{2\sqrt{1-\frac{x^2}{2}}}}dx$$