It's a bit more subtle than that. What saturates is the actuator. You could choose a single minimum/maximum value for the integrator, but it could be better ye

It's a bit more subtle than that. What saturates is the actuator. You could choose a single minimum/maximum value for the integrator, but it could be better ye to have a value that depends dynamically on how much room there is left in the actuator output. For example, if the actuator is saturated, the integral term should not be increasing. That could happen at different values of the integral, so there is not a single min/max.