The purpose of R2 is unclear without detailed informations about the gyro characteristic. Or it's just an erroneous or wrongly copied schematic.
I would expect a reset means for the integrator.
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Still wondering about the gyro output impedance. Without this information, the discussion is meaningless I fear.
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In the first case, R2 would be connected between tow low impedance and haven't any effect on circuit behaviour, in the second case, R2 would provide feedback.
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No, in case of a current source the circuit effectively acts as I/V converter.
In both cases, the design is questionable in my view. But It's generally problematic if you neither knew the intended purpose nor the circuit parameters.