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For a uniform partition of the radius $[0, R]$ into $n$ subintervals, the difference between the upper sum (overestimation) and the lower sum (underestimation) is exactly $\frac{2\pi R^2}{n}$. This demonstrates that the error decreases inversely with the number of partitions $n$.

Conditions: Uniform partition of radius $R$ into $n$ steps; Linear circumference function $C(r) = 2\pi r$

For a uniform partition of the radius $[0, R]$ into $n$ subintervals, the difference between the upper sum (overestimation) and the lower sum (underestimation) is exactly $\frac{2\pi R^2}{n}$. This demonstrates that the error decreases inversely with the number of partitions $n$.

Conditions: Uniform partition of radius $R$ into $n$ steps; Linear circumference function $C(r) = 2\pi r$