How to calculate net force from a Lorentz Force Density in Magnets
My intuition was adding all the values in a face, and multiplying it with the area of the face will give the force in that face


Yes, basically that's what you have to do.
The mathematical term for that is integrating the force densisty over the area and about since Leibniz's time it is written as 

\$F = \int_A f da\$  

(where \$f\$ is force density [unit force per unit area]; \$da\$ is the infinitesimal area element; \$A\$ is the total area and \$F\$ is the total force)
BTW: the \$\int\$ is to remind of Sum.