This is part of a homework problem. I want to actually solve it myself, so no solutions, please (although this isn't even the full problem statement). I don't have a very good grasp on differential forms and pullbacks though, so I need to make sure I'm on the right track.
Let $A \subseteq \mathbb{R^2}$, $\alpha : A \rightarrow \mathbb{R^3}$ be defined by $\alpha(u,v) = (u,v,u^2+v^2+1)$, and $Y = \alpha(A)$. I want to compute $\int_{Y_\alpha}\omega$, where $\omega = x_2dx_2\wedge dx_3$.
Computing $\alpha^* \omega$ gives me $v dv \wedge du$. Then $\int_{Y_\alpha}\omega = \int_A \alpha^* \omega = \int_A v dv\wedge du = \int_A v\ dvdu$ and I just evaluate it as a two-dimensional integral.
Am I on the right track?
Well, $dx_2 = dv$ and $dx_3 =2udu + 2vdv$, so what is $dx_2 \wedge dx_3$?