Generally Integeration means sum(adding) or total...
so for example,
say to calculate the mass flow rate across a cross section, where density, velocity is not uniform in that cross section.. then you can calculate the mass flow rate in a small region dA as = velocity * density * dA
so, So the total mass flow rate across a given area can be computed by adding the mass flow across many such small elemental areas(dA). so to represent this in mathametics we can write integration ( velocity * denstiy * dA) if we know the variation of velocity and density in that area then you can compute the mass flow rate as integration ( velocity * denstiy * dA) where limits of the integration will be your area of interst across which you want to caluclate the total mass flow rate