The Maxwell Stress is simply:
\$ \sigma = \frac{B^2}{2 \mu_0} \$
Which, you'll notice, is also a unit of energy density (which, multiplied by the volume the field occupies, this gives the total electrical energy stored in the field).
If you have B applied to one end of a magnetic armature and zero to the other end, then this is the pressure applied to the armature. Multiplied by the area of the pole pieces (that carry the magnetic field), this gives the force.
For example, a 12mm dia. armature at 1.5T (typically the limit for steel pole pieces), with no field leakage, would support about 10kgf.
As Ammar hinted, this generalizes to the Lorentz force and whatnot. But if you can make extreme simplifications, you don't need to integrate vectors at all. Use geometry and physics to solve them for you!
Tim