I would like to approximate the following integral of a product:
$$ I = \int dz\, f(z)\prod_{i=1}^n\left(1 - \rho_i(z)\right) $$
The functions $f$ and $\rho_i$ are differentiable for all $i$, $\rho_i(z) \ll 1$ for all $z$ and $i$, and $\int dz\, f(z)=1$.
My initial hunch was to exchange the order of the integral and the product because the cross-terms of $\rho$s are likely small. Then
$$ I \approx \prod_{i=1}^n\left(1- \int dz\, f(z)\rho_i(z)\right). $$
Are there better and/or more well-motivated approximations?