Lorentz force of a conductor loop
A bad misconception! In homogenic magnetic field you can move the whole cross product and the field density vector outside the integral, you integrate only the differential of the position vector. That's zero around a closed loop, there's no cumulated force to your current loop. Parts of the loop surely get some force and there can well be a torque, as we have in electric motors.

I guess your actual task will remain unsolved, if you build on the assumption that a closed current loop catches a non-zero resultant force in a homogenic magnetic field. Maybe you should reveal the task. If the math problem that you presented is the exact task you have got, it's a prank.