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For any $n geq 2$, let
$$D_n [r , (a_1, b_1 ) , ldots , (a_n, b_n) ] =
{ (x_1 , ldots , x_n ) in mathbb R^n mid
sum_i x_i = r mbox{ and } b_i geq x_i geq a_i , forall i },$$
where $r geq b_i geq a_i geq 0$ for all $i$, and all are real numbers.

Question: What is the 'volume' of $D_n [r , (a_1, b_1 ) , ldots , (a_n, b_n) ]$?

So for example for $n=2$, this set is either empty or it is some short line, and the purpose would be to calculate the length of that line (in terms of the parameters $r, a_i, b_i$). This is easy, I've done that already. Also the case $n=3$ would in principle still be doable to do by hand: it would be either zero (in case the set is empty) or part of a plane in $mathbb R^3$, the area of which we desire to compute.

Now to come up with a formula for the case $n=3$ (and higher) in a smarter way, my idea was to reason inductively from the case $n=2$, so basically reducing the three dimensional case to the two dimensional one, etc. I obviously tried to use integrals.

The problem I run into, is that in order to compute the volume we want with integrals, we have to view this set as (a subset of) an $n-1$-dimensional space. (Indeed, the integral of the constant function $1$ over the region $D_n [r , (a_1, b_1 ) , ldots , (a_n, b_n) ]$ seen as part of $mathbb R^n$ (rather than $mathbb R^{n-1}$), is equal to $0$. That's not what we want.) But to do that, it seems to me that we would need a concrete isometric embedding of $D_n [r , (a_1, b_1 ) , ldots , (a_n, b_n) ]$ into $mathbb R^{n-1}$, and I can't really find a nice one.

Do you have an idea about how to approach this problem best?

(This question was previously posted on Math.StackExchange, see https://math.stackexchange.com/questions/3135606/computing-hyper-area-of-a-contrained-simplex.)

Computing volumes of polytopes in general is NP-hard. However, your polytope is special - it is a slice of a hypercube by a hyperplane, and this is tractable, see Theorem 1 in:

Marichal, Jean-Luc; Mossinghoff, Michael J., Slices, slabs, and sections of the unit hypercube, Online J. Anal. Comb. 3, Article 1, 11 p. (2008). ZBL1189.52011.

The $(n-1)$-volume of your polytope (in $mathbb R^n$) equals the $(n-1)$-volume of the polytope
begin{multline}
P:={(x_1,dots,x_{n-1})inmathbb R^{n-1}colon \
a_ile x_ile b_i forall i=1,dots,n-1,\
a_nle r-sum_1^{n-1}x_ile b_n}
end{multline}
(in $mathbb R^{n-1}$) divided by $1/sqrt n$, which latter is the cosine of the angle between the unit vectors $(1/sqrt n,dots,1/sqrt n)$ and $(0,dots,0,1)$ in $mathbb R^n$ -- because $P$ is the image of your polytope under the orthogonal projection of $mathbb R^n$ onto $mathbb R^{n-1}$ given by $(x_1,dots,x_n)mapsto(x_1,dots,x_{n-1})$. The unit vectors $(1/sqrt n,dots,1/sqrt n)$ and $(0,dots,0,1)$ are normal vectors to, respectively, the hyperplane containing your polytope and the hyperplane ${(x_1,dots,x_n)inmathbb R^ncolon x_n=0}$; the latter hyperplane is identified with $mathbb R^{n-1}$.

A formula for the volume of a polytope was given by Lawrence.

We may assume without loss of generality that $a_i=0$. If
$r$ and each $b_i$ are positive integers, then consider
$$ f(x) = frac{left( 1-x^{tb_1+1}right)cdots
left( 1-x^{tb_n+1}right)}{(1-x)^n}. $$
The coefficient of $x^{tr}$ is a polynomial function of $t$,
and the volume $V$ will be its leading coefficient. If I didn't
make a computational error, then
$$ V=frac{1}{(n-1)!}sum_{substack{Ssubseteq
{1,dots,n}\ sum_{iin S}b_i<r}} (-1)^{|S|}
left(r-sum_{iin S}b_iright)^{n-1}. $$
If this isn't correct, then something close to it will be.