This question received no answers on [CS.SE][1], and I thought MO (not M.SE nor CSTheory.SE) was the right place to ask, as it involves both integrals and complexity.

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 $J$. 

  


  [1]: http://cs.stackexchange.com/questions/12896/a-hard-n-fold-integral