What numerical method can approximately compute the $(n-1)$-dimensional surface area of the $\ell_p$ ball $\{x\in\mathbb R^n: \sum_{i=1}^n |x_i|^p=1\}$, for $p\in[1,\infty)$? Ideally the method should handle $n$ and $p$ both in the range of 5 to 10.

One approach begins with the definition of surface area as $$ \lim_{\varepsilon\to 0^+} \frac{\mu_n(B_p + \varepsilon B_2)-\mu_n(B_p)}{\varepsilon}, $$ where $B_p$ and $B_2$ are unit $\ell_p$ and $\ell_2$ balls, using Monte Carlo to estimate the volume of both bodies. This method fails numerically, because when $\varepsilon\to 0^+$ the two volumes are very close to each other.

Another approach uses *Cauchy's integration formula*, which states that
the volume of $\partial B_p$ is equal to
$$
\frac{1}{\mu_{n-1}(B_2)}\int_{S_2}\mu_{n-1}(B _p|u) du,
$$
where $\mu_{n-1}(B_p|u)$ is the volume of $B_p$ projecting onto the orthogonal complement of $u$. However, this projection seems difficult to numerically approximate.

What approaches would provide a better approximation?