can isolate the divergence for small $a$ in the integral of (\[strip area\]) as follows $$\begin{aligned}
\label{split 1}
A_d(z_{\textrm{max}},a) &=&
\int_a^{z_{\textrm{max}}} 
\frac{2}{w^d \sqrt{f(w)}}\,dw
+
\int_a^{z_{\textrm{max}}} 
\frac{2}{w^d \sqrt{f(w)}}\left(\frac{z_{\textrm{max}}^d}{\sqrt{z_{\textrm{max}}^{2d}-w^{2d}}}-1\right) dw\,
\hspace{1cm}\\
\label{I def}
\rule{0pt}{.7cm}
&\equiv&
\int_a^{z_{\textrm{max}}} 
\frac{2}{w^d \sqrt{f(w)}}\,dw + I_d(a, z_{\textrm{max}})\;.\end{aligned}$$ The finite term in the expansion for small $a$ is now given by $I_d(0, z_{\textrm{max}})$ plus a contribution from the first integral in (\[I def\]). The distinction between (\[split 0\]) and (\[split 1\]) is obviously meaningless for $AdS_{d+2}$, where $f(z)=1$ identically. The splitting (\[split 1\]) has been used in the appendix \[app more on nh\] to get some insights about the expansion of the finite term of the minimal area as a power series as $z_0- z_{\textrm{max}}$ and the possibility to approximate it through the near horizon geometry.

A more general ansatz {#more general ansatz section}
---------------------

In this section we consider a more complicated expression for the metric in order to include other kind of black holes in our discussion. Let us take a $D+1$ dimensional spacetime and the following ansatz for the metric on the constant time slice $$\label{ds2 general ansatz}
ds_0^2\,=\,\left(\frac{dr^2}{A(r)^2}+B(r)^2 d\vec{x}^2\right)e^{-\frac{D-1-d}{d}\chi(r)}
+R^2 e^{\chi(r)} \gamma^{\textrm{(c)}}_{ij}(r) d\theta^i \theta^j$$ where $d\vec{x}^2$ gives the metric of $\mathbb{R}^d$ and $\gamma^{\textrm{(c)}}_{ij}(r)$ is the 