I'm looking to estimate the definite integral,
$$\int_0^\frac{1}{\sqrt{n-1}}\int_{x_1}^\frac{1}{\sqrt{n-2}}\cdots\int_{x_{n-2}}^\frac{1}{\sqrt{1}}dx_{n-1}\cdots dx_{2}dx_1.$$
Obviously, one can show that this is less than $\frac{1}{\sqrt{(n-1)!}}$. I suspect that this estimate can be greatly improved. Are there any standard approaches to a problem such as this? A hint would be appreciated!
Kind of hand-wavy: This is an $n$-dimensional cone, so its volume is $1/n$ times the volume of the corresponding cylinder which would be the same iterated integral with all the lower bounds changed to $0$. So I think the volume is $\frac{1}{n\sqrt{n!}}.$