What is the specific error magnitude between the upper and lower Riemann sums for a uniform partition of the circle's area?

Conditions

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

Reasoning, step by step

1. Define the width of each subinterval as $\Delta r = R/n$.
2. Identify that the upper sum uses the right endpoint (larger circumference) and the lower sum uses the left endpoint (smaller circumference) for each strip.
3. Note that for a linear function, the difference between consecutive max and min heights in a strip is constant relative to the slope.
4. Calculate the area of the 'excess' triangles or rectangles formed by the gap between upper and lower sums.
5. Sum these gaps across all $n$ intervals.
6. Derive the final expression: Total Error $= n \times (\text{slope} \times \Delta r) \times \Delta r / 2$? No, simpler view: The total vertical span covered by the difference is effectively the range of the function times the width? Actually, for monotonic functions, Upper - Lower $= (f(R)-f(0))\Delta r$. Here $f(R)=2\pi R, f(0)=0$. So Error $= 2\pi R \cdot (R/n) = 2\pi R^2/n$.

Example

The script states: 'On a uniform partition, the upper and lower sums differ by 2πR²/n. The error tends to zero...'

Common misconceptions

• Assuming the error depends on the square of $n$.
• Confusing the absolute error with the relative error percentage.

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