Stuck in trace theorem

Question

I am reading Sobolev space in the book Partial Differential Equation by Evan and I do not understand some point in the proof of the trace theorem.
Let $U$ is open, bounded and $\partial U$ is $C^1$. Let $x_0\in\partial U$ and assume that $\partial U$ is flat near $x_0$ lying in the plane $\{x_n=0\}$. Choose a ball $B(x_0,r)$ such that $B^+=B\cap \{x_n\ge 0\}\subset U$ and $B^-=B\cap \{x_n\le 0\}\subset \mathbb{R}^n-U$. Let $\hat{B}=B(x_0,r/2)$. Choose $\zeta\in C_c^{\infty}(B)$, with $\zeta=1$ on $\hat{B}$. Define $\Gamma=\hat{B}\cap\partial U$. Set $x'=(x_1,\ldots,x_{n-1})\in\mathbb{R}^{n-1}=\{x_n=0\}$. The author then claims that
$$\int_{\Gamma}|u|^pdx'\le\int_{x_n=0}\zeta|u|^pdx'=-\int_{B^+}(\zeta|u|^p)_{x_n}dx.$$ What is the meaning of $\int_{\Gamma}|u|^pdx'$ ? How do we get the equality ?