\int_a^{z_{\textrm{max}}} 
\frac{z_{\textrm{max}}^d}{w^d \sqrt{f(w)}\sqrt{z_{\textrm{max}}^{2d}-w^{2d}}}\,dw
\,\equiv\,
R^dL_\perp^{d-1} A_d(z_{\textrm{max}},a)\:.$$ In order to isolate the divergence of (\[strip area\]) as $a \rightarrow 0$, we write the integral as follows $$\begin{aligned}
\label{split 0}
A_d(z_{\textrm{max}},a) &=&
\int_a^{z_{\textrm{max}}} 
\frac{2}{w^d}\,dw+
\int_a^{z_{\textrm{max}}} 
\frac{2}{w^d}\left(\frac{z_{\textrm{max}}^d}{\sqrt{f(w)}\,\sqrt{z_{\textrm{max}}^{2d}-w^{2d}}}-1\right) dw
\\
\label{split 0bis}
\rule{0pt}{.7cm}
& \equiv & 
\frac{2}{(d-1)\,a^{d-1}}
+ \mathcal{A}_d(z_{\textrm{max}},a)\;.\end{aligned}$$ The second integral in (\[split 0\]) is finite as $a \rightarrow 0$ because we have either $f(w)=1+O(w^{d+1})$ or $f(w)=1+O(w^{d})$ for $w \rightarrow 0$ (see (\[charged BH metric z\]) and (\[lifshitz metric\]) respectively). In (\[split 0bis\]) we have introduced the finite term in the UV cutoff expansion $$\label{calArea}
\mathcal{A}_d(z_{\textrm{max}},a) 
\,\equiv\,
\int_a^{z_{\textrm{max}}} 
\frac{2}{w^d}\left(\frac{z_{\textrm{max}}^d}{\sqrt{f(w)}\,\sqrt{z_{\textrm{max}}^{2d}-w^{2d}}}-1\right) dw
-\frac{2}{(d-1)z_{\textrm{max}}^{d-1}}\;.$$ In this paper we will be mainly interested $O(1)$ term in the $a$ expansion, which is $\mathcal{A}_d(z_{\textrm{max}},0)$. In the figure \[plotA0\] this term is shown for the charged black hole in $AdS_4$ at zero temperature. As the turning point $z_{\textrm{max}}$ approaches the horizon $z_0$, it develops a divergence we are going to study.

![Charged black hole, extremal case and $z_0=1$. Plot of the finite term $\mathcal{A}_2(z_{\textrm{max}},0)$ as function of $z_{\textrm{max}}$ (red curve). Close to the boundary (i.e. when $z_{\textrm{max}} \rightarrow 0$) it coincides with the curve corresponding to $AdS_4$ (blue curve), which can be read from (\[A expansion empty\]). \[plotA0\]](plotA0){width="10cm"}

Equivalently, we 