Can integrals be interpreted simultaneously as aggregates and averages? (Mas-Colell et al. 1995, Proposition 4.C.4)
I talked to my teacher to try and get a handle on the question.

You are correct that usually

$$\overline {f(x)} = \frac{1}{b-a}\int^b_a f(x) dx$$

But for the demand function there are two arguments, $w$ and $p$.

$$x(p) = \int_0^\bar W \tilde x(p,w) dw$$


The left hand side is no longer a function of $w$. The integral is to simplify wealth out of the picture, but we aren't averaging over wealth.
All consumer preferences are identical. For the sake of argument, suppose there were $n$ consumers. We aren't multiplying by $n$ on the inside of the integral, so we don't need to divide out anything (like $n$) through the outside of the integral. The "aggregation" is occurring over all consumers.