This is a generator in a conductive environment. It says in my textbook that the work of the electric force in the conductive environment (outside the generator), where the charge is only driven by the electrical force and not the electromotive one, is equal to \int_{a}^{b}\vec{F_e}\mathrm{d}\vec{l} = Q\int_{a}^{b}\vec{E}\mathrm{d}\vec{l}=-Q(V_b-V_a) > 0

I don't get it how this integration worked. If the electric field and the line segment are collinear with the same direction outside the generator.