How to rewrite the surface integral $\int_{(s_1,\ldots,s_n) \in \mathbb S_{n-1}}\sqrt{1-s_1^2}ds$ as a special function

Question

Let $n \ge 2$ be an integer and let $\mathbb S_{n-1}$ be the $(n-1)$-dimensional unit-sphere.

Question. How to rewrite the surface integral $I_n:= \int_{(s_1,\ldots,s_n) \in \mathbb S_{n-1}}\sqrt{1-s_1^2}ds$ as a special function ?

Note that $I_n = 2\sigma_nh_n$, where $\sigma_n$ is the surface area of $\mathbb S_{n-1}$ and $h_n$ is the average "co-height" of a point a hemisphere, sampled according to the uniform measure on $\mathbb S_{n-1}$.