Integration over manifold

Let $M$ be a smooth 2-manifold in $\mathbb{R}^3$ such that $$4x^2+y^2+4z^2 = 4, y \ge 0$$
The boundary of $M$ is the set of points where $$x^2 + z^2 = 1, y = 0$$

Let $\alpha(u,v) = (u,2\sqrt{(1-u^2-v^2)},v)$ for $u^2+v^2 = 1$ (open unit disc)

Let $\omega = x_2dx_1+3x_1dx_3$

The goal of the problem is to verify Stoke's Theorem on this example

But I'm having trouble evaluating

$\int_{M}d\omega$ and $\int_{\partial M}\omega$

I know that I need to take the pull back of $\alpha$. so pretty much integrate $\alpha^*(\omega)$ over the unit disc and the boundary to obtain the answer to the integrals above. But I'm having trouble doing that.

• perhaps $x_1=x; x_2=y, x_3=z$ ? – Thomas Apr 7 '16 at 12:25