I am trying to integrate the 2-form
$$\eta=\frac{1}{\|\mathbb{x}\|^m}(x_1dx_2\wedge dx_3-x_2dx_1\wedge dx_3+x_3dx_1\wedge dx_2)$$ over $Y_\alpha$ where $$\alpha(u, v)=(u, v, (1-u^2-v^2)^{1/2})$$ and $Y=\alpha(S^1)$.
I did $$\int_{Y_\alpha}\eta= \int_Au\det (\frac{\partial \alpha_{2 3}}{\partial (u, v)})-\int_Av\det (\frac{\partial \alpha_{13}}{\partial (u, v)})+\int_A(1-u^2-v^2)^{1/2}\det (\frac{\partial \alpha_{12}}{\partial (u, v)})$$
but I can't find a way to integrate the expressions beyond this, so I must be doing something wrong.
Any hints?