CES preferences intuition I asked this question on math.stackexchange but deleted it from there and brought it here.

I had a question about Constant Elasticity of substitution type preferences of the form: $$U=\int_{0}^{1}(c(\omega)^{\rho}d\omega)^{\frac{1}{\rho}}$$
 where the paramter $\rho$
  governs the degree of substitutability between goods. Here, $c(\omega)$
  represents consumption of good $\omega$
  which exists on the unit interval. As such, this type of preference specification aggregates over consumption of different goods.

My question is as follows. I have always thought of integrals of the form: $$I=\int f(x)dx$$
  as approximating sums of areas of infinitesimally small rectangles (in terms of their base) and heights being determined by $f(x)$
 . In the case of the example above, what really is $c(\omega)$
 ? Is it a function?
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.