Evaluate $\int_{0<x_1,\cdots,x_n<1,\ 0<(x_1\cdots x_n)^{\frac{1}{n}}<a}dx_1\cdots dx_n$

Question

$1$. How to prove that for $n\in\mathbb{N}, a\in(0,1)$ one have
$$f(a,0):=\int_{0<x_1,\cdots,x_n<1,\ 0<(x_1\cdots x_n)^{\frac{1}{n}}<a}dx_1\cdots dx_n=a^n \sum_{k-0}^{n-1}\frac{(-n\log(a))^k}{k!}$$
This identity arises from probability theory, but I wonder if it's solvable using calculus alone.

$2$. Moreover, for $p\in \mathbb{R}$, can we give a closed form to the generalized
$$f(a,p):=\int_{0<x_1<1,\ \cdots,\ 0<x_n<1,\ 0<\left(\frac1n \sum _{i=1}^n x_i^p\right)^\frac{1}{p}<a}dx_1\cdots dx_n$$
This one is rather open. Thanks in advance!

Answer 1

1. Let $f_n(a)=\idotsint\limits_{\substack{0<x_1,\ldots,x_n<1\\x_1\cdots x_n<a}}dx_1\cdots dx_n$ for $a>0$ (for $a<1$ it is your $f(a^{1/n},0)$; for $a\geqslant 1$ it is $1$). Then $$x_n=x\implies(x_1\cdots x_n<a\iff x_1\cdots x_{n-1}<a/x),$$ thus $f_n(a)=\int_0^1 f_{n-1}(a/x)\,dx=a+\int_a^1 f_{n-1}(a/x)\,dx$ for $0<a<1$. (Now induction works.)
2. This time, for $g_n(a)=\idotsint\limits_{\substack{0<x_1,\ldots,x_n<1\\x_1^p+\ldots+x_n^p<a}}dx_1\cdots dx_n$ (and $p>0$), we get $g_1(a)=\min\{a^{1/p},1\}$ and $$g_n(a)=\int_0^{a}g_{n-1}(a-x^p)\,dx,$$ which doesn't look explicitly solvable ($n=2$ already looks clumsy).