What would you like to understand?

Unrolling a curved ring into a straight strip creates an approximation because finite thickness causes slight curvature mismatch. The error is not eliminated immediately but must be controlled: as the partition width $\Delta r$ tends to zero, the total accumulated error vanishes, allowing the sum to converge to the exact area.

Conditions: The ring has finite width $\Delta r$; The partition is being refined ($\Delta r \to 0$)

Unrolling a curved ring into a straight strip creates an approximation because finite thickness causes slight curvature mismatch. The error is not eliminated immediately but must be controlled: as the partition width $\Delta r$ tends to zero, the total accumulated error vanishes, allowing the sum to converge to the exact area.

Conditions: The ring has finite width $\Delta r$; The partition is being refined ($\Delta r \to 0$)