CES preferences intuition
It is my impression that the correct expression for "Dixit-Stiglitz" preferences is 

$$U=\left(\int_{0}^{1}c(\omega)^{\rho}d\omega\right)^{\frac{1}{\rho}}$$

which then can be seen as a continuous incarnation (in [0,1]) of, say

$$\left (\sum_{i=1}^na_i\omega_i^{\rho}\right)^{\frac{1}{\rho}}$$

with $c(\omega_i) = a_i^{1/\rho}\omega_i$.

In other words, a definite (Riemann) integral is indeed conceived as a sum of infinitesimally small rectangles, but it can also be seen as the continuous incarnation of a sum.

A formal link between an integral and a sum is provided by the Euler-MacLaurin formula.