Integration over a set in n-dimensional space

[sphere1]

Homework Statement

Not a homework problem, but self-studying Example 1.2.24 from Hogg et al.'s Introduction to Mathematical Statistics (7th ed).

Relevant Equations

N/A

Example 1.2.24: Let C be a set in n-dimensional space, and let Q(C) = ∫...∫_Cdx₁dx₂...dx_n. If C= {(x₁,x₂,...,x_n : 0 ≤ x₁ ≤ x₂ ≤ ... ≤ x_n ≤ 1}, then Q(C) = ∫₀¹ ∫₀^x_n∫₀^x_n-1...∫₀^x₃∫₀^x₂dx₁dx₂...dx_n = 1/n!. (Apologies for the poor formatting of the integration bounds.)

I haven't the foggiest idea how to get to 1/n!. My solution for the latter integral is x₂x₃...x_n. Even if I were to assume that [0, 1] were divided into intervals of equal length, giving x₁ = 1/n, x₂ = 2/n, etc., I get n!/nⁿ, which I'm pretty sure doesn't reduce to 1/n!. My assumption is that I am somehow integrating the second integral incorrectly, but I'm not sure how. Any suggestions would be appreciated!

 

[Orodruin]

I suggest you start by solving the more general integral
$$
I_n(x) = \int_0^{x} \int_0^{x_{n}} \cdot \int_0^{x_2} dx_1\, dx_2\ldots dx_{n}.
$$
You can solve this by induction. The base case is $$I_1(x) = \int_0^x dx_1 = x$$.

Compute the integral $I_n(x)$ for the first few values of $n$ and try to see the pattern. Then show that it is the correct pattern by induction applying$$I_{n+1}(x) = \int_0^x I_n(x_{n+1}) dx_{n+1}$$