Given $x\in\mathbb{R}^n$, $x_i$ denotes its $i$-th coordinate. My question is:

-  What is $Vol(\{x\in[0,1]^n|\sum_{i=1}^n x_i\le t\})$ for $t\in\mathbb{R}$ ? Is there some kind of "easily computable" formula for it ?

A colleague asked me this in relation to a problem in applied mathematics, and I thought that the answer was going to be easy to find, but after trying to compute it by hand in dimension $3$ and then generalizing, I just surrendered. I am pretty sure that it should be somehow classical, but I could not track down a formula or a reference in the general case. I should specify that I know next to nothing about section of convex bodies, and I might be missing something easy. With some browsing, I found the following partial answers (which I hope I do not mix-up):

- If $t$ is an integer, then the volume is the sum of the $t$ first Eulerian numbers divided by $n!$. 

- Between and integer $t<n$ and the next, the volume function is a polynomial of degree $n$. The polynomials corresponding to two consecutive intervals must have the same first derivative at their common point. For instance, for $n=2$ the volume is given by $\frac{t^2}{2}$ if $t\in[0,1]$, $1 - \frac{(2-t)^2}{2}$ for $t\in[1,2]$ and constant elsewhere.

- There is a Fourier-analysis approach for the problem of finding the $n-1$-volume of the section of a convex body by an hyperplane, but the formulas I found did not seem to me easily computable (my knowledge of Fourier analysis is approximatively equivalent to that of convex bodies, I must say).

- We can use some probability theory to find an approximation: each $x_i$ is seen as a random variable following a law of mean $\mu$ and variance $\nu$, then $S=\sum x_i$ follows (approximatively  and for large $n$) the normal law with mean $n\mu$ and variance $n^2\nu$, and the probability of $S\le t$ is then easy to compute. I presume that the meaning of "$n$ is large" and "approximate" can be made explicit by looking more closely at the central limit theorem. (This idea comes from Michael Lugo's answer to <a href="http://mathoverflow.net/questions/22648/area-of-cross-section-at-midpoint-perpendicular-to-longest-diagonal-in-the-uni/22670#22670">this related question</a>. Another answer by Andrey Rekalo points to the Fourier-analysis method mentioned above.) 

The last point brings a more general question: given a probability distribution on $[0,1]$ (perhaps in some well behaved class), are there some good estimates for $P(\sum_{i=1}^n x_i\le t)$ for smaller $n$ ?