Add option to integrate filter to divide by volume

My user wants to add a feature to the integrate filter that allows not just the sum of the items, but sum or integral divided by the total volume.

From my user:

Can we get a version that provides the average of the quantity? That is, currently if you look at variable f(x) (x \in 3D), we are getting \int_D f(x) dx -- an integral of f over the domain D. However, we often want the average of f over the domain, (\int_D f(x) dx) / (\int_D dx). Setting up a new variable to scale by the volume for every variable and every plot is annoying...