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The argument places strips of height $2\pi r$ along the radius axis, forming a Riemann sum. As the maximum partition width tends to zero, these sums approach the area beneath the linear circumference function.

Conditions: The partition is refined such that the maximum width tends to zero.; The circumference function $2\pi r$ is treated as the height of the strips.

The argument places strips of height $2\pi r$ along the radius axis, forming a Riemann sum. As the maximum partition width tends to zero, these sums approach the area beneath the linear circumference function.

Conditions: The partition is refined such that the maximum width tends to zero.; The circumference function $2\pi r$ is treated as the height of the strips.

By exploiting radial symmetry, the disk is decomposed into concentric rings. The variable $r$ represents the distance from the center to an inner boundary (a radius, not a diameter).

Conditions: The shape being analyzed is a circle or disk.; The decomposition utilizes radial symmetry.

By exploiting radial symmetry, the disk is decomposed into concentric rings. The variable $r$ represents the distance from the center to an inner boundary (a radius, not a diameter).

Conditions: The shape being analyzed is a circle or disk.; The decomposition utilizes radial symmetry.

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 ($\Delta r \to 0$)

Unrolling a curved ring into a rectangle is an approximation because the ring has finite thickness. A curved strip of finite width is not exactly identical to a straight rectangle; the outer and inner circumferences differ.

Conditions: The ring has a finite width $\Delta r$.; The approximation replaces a curved annulus with a straight rectangle.

Unrolling a curved ring into a rectangle is an approximation because the ring has finite thickness. A curved strip of finite width is not exactly identical to a straight rectangle; the outer and inner circumferences differ.

Conditions: The ring has a finite width $\Delta r$.; The approximation replaces a curved annulus with a straight rectangle.