Integration of Area inside an Ode's

Dear collegues,
for my study, a shape of the meniscus of a molten metal due to lorentz' forces, i coupled the moving mesh ale and an ode module.
The ode is needed to guarantee the moving mesh ale the conservation of mass: the result of the ode should be the maximum height of the meniscus, an information that will be spread on all boundaries where i applied a force displacement.

Can anyone help me, how to define an area integration with an ode?

Regards
Gianmarco