Computing sieve integrals using LattE, and the density of integers with a localized divisor

We consider the problem of estimating numerically integrals of the shape $$ \int_P \frac{dt}{t_1 \dotsb t_k} $$ where $P \in {\mathbb R}_{>0}^k$ is a convex polytope, $t=(t_1,\dotsc, t_k)$ and $d t$ is the Lebesgue measure. This type of integral appears frequently in main terms of sieve theory.
We propose a simple method, based on the LattE software for integration of polynomials over polytopes, which computes rigorous bounds on this integral in polynomial time with respect to the precision (in bits). We test the method on several examples from the literature of sieve theory.
We apply our results to compute numerical approximations to the natural density $$ h(\alpha, \beta) := \operatorname{density}\{n\in{\mathbb N}, \exists d\mid n, d\in [n^\alpha, n^\beta]\}, \qquad (0<\alpha<\beta<1) $$ of integers having a localized divisor, in the region $\beta - \alpha \geq 0.02$. One ingredient involved is a refined formula for $h(\alpha, \beta)$ which involves a manageable number of terms for these $\alpha, \beta$. As a corollary, we give a numerical approximation of the leading constant in a theorem of Haddad and Koukoulopoulos on the average of the logarithm of middle-divisors of integers.