A hard $n$-fold integral Consider the $n$-fold integral 
$$
J = \int_{\theta_1 \in I_1, \theta_2 \in I_2 \ldots, \theta_n \in I_n} d\theta_n\ldots d\theta_2 d\theta_1
$$

whose intervals are defined by 
$$
\begin{align}
I_1 = [0,1] \\
I_i = [\max(c_i,\theta_{i-1}),1] , 2\leq i\leq n  
\end{align}
$$

and the $c_i \in [0,1]$ are predefined rational constants. Given a rational $v\in [0,1]$, 


  is deciding if $J=v$ NP-hard? 


Informally , each $\max$ in the lower limits of the intervals leads to a two-way split in evaluating the integral, and thus to $2^{n-1}$ integrals that sum to $J$.
Your problem is a special case of the following: given a convex polyhedron defined by a set of linear inequalities, compute its volume.  I believe that problem has been studied in the literature.

For instance, for related work on computing the volume of convex polyhedra/polytopes, see the following:


Algorithm for finding the volume of a convex polytope
Computing the Volume of a Convex Polytope Using Interval Arithmetic
Polytope volume computation


There is probably lots more work out there that might be relevant, but maybe this gives you a start.  You might start by reading that work to see if there are any techniques you can adapt; and by doing a literature review with the above as an entry point into the literature.