By general theory, what the statement really means is that for any singular point $x$ in $W$, there is a neighborhood $U$, and a nowhere zero holomorphic $n$ form $\Theta$ on $W^{reg}\cap U$ with $\int_{W^{reg}\cap U}\Theta\wedge\overline{\Theta}<\infty$. We first consider the cases $L=K_{X}^{2}$ and $K_{X}^{-1}$. Previous discussion has shown that for any $x$, there is a neighborhood $U$ of $x$, an integer $k>0$, a constant $C>0$, and a section $s$ of $L^{k}$ over $X_{\infty}\setminus\Sigma=W^{reg}$ with $C^{-1}\leq\|s(x)\|^{2}\leq C$ for $x\in W^{reg}\cap U$. Here the norm is taken with respect to the Kähler-Einstein metric. When $L=K_{X}^{2}$, we define $\Theta=(s\otimes\overline{s})^{\frac{1}{2k}}$, then

$$\int_{W^{reg}\cap U}\Theta\wedge\overline{\Theta}=\int_{W^{reg}\cap U}\|s\|^{\frac{1}{k}}dvol\leq C^{\frac{1}{2k}}Vol(W).$$

When $L=-K_{X}$, we define $\Theta=(s^{*}\otimes\overline{s^{*}})^{\frac{1}{k}}$, where $s^{*}$ is the dual section of $s$. So $\|s^{*}\|=\|s\|^{-1}$. Then

$$\int_{W^{reg}\cap U}\Theta\wedge\overline{\Theta}=\int_{W^{reg}\cap U}\|s^{*}\|^{\frac{2}{k}}dvol\leq C^{-\frac{1}{k}}Vol(W).$$

In the Calabi-Yau case, since $\Theta_{i}$ has norm one, we easily see that there is a limit holomorphic volume form $\Theta$ on $X_{\infty}\setminus\Sigma=W^{reg}$ with norm one. Then $\int_{W^{reg}}\Theta\wedge\overline{\Theta}=Vol(W).$ ∎