Strange coarea formula on the cylinder. Is it correct?

Question

Consider the cylinder $[-1,1]\times S^{1}$, where $S^{1}=\mathbb{R} / \mathbb{Z}$ and let $x$ be the coordinate on $[-1,1]$ and $y$ the coordinate on $S^1$. Let $\alpha$ be a 1-form on $[-1,1]\times S^{1}$. Is the following expression correct: $\int_{[-1,1]\times S^1}\alpha \wedge dx = \int_{-1}^{1}\left ( \int_{\{x\}\times S^1}\alpha \right )dx$ ? Is there any minus sign or constant missing?

Answer 1

This is just Fubini's Theorem. However, yes, you are off by a minus sign. The "standard" area 2-form on $[-1,1]\times S^1$ will be $dx\wedge d\theta$, and so you should have your trial 2-form in the order $dx\wedge \alpha$ in order to get your formula. (Just write $\alpha = f\,dx + g\,d\theta$ to see this.)