Chapter 6 — Calculus II
Lesson map
Lesson 06 published. Map, example, EX016.
flowchart TD
N0["Learning outcomes"]
N1["6.1 From flow to volume"]
N2["6.2 Numerical error"]
N3["6.3 Accumulation and reconciliation"]
N4["6.4 Series for forecasting"]
N0 --> N1 --> N2 --> N3 --> N4
N4 --> EX["Worked example"]
EX --> Q["Exercise EX016"]
Logical drawing
flowchart LR
N0["Learning outcomes"]
N1["6.1 From flow to volume"]
N2["6.2 Numerical error"]
N3["6.3 Accumulation and reconciliation"]
N4["6.4 Series for forecasting"]
N0 --> N1 --> N2 --> N3 --> N4 --> Q["EX016"]
[Learning outcomes]
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v
[6.1 From flow to volume]
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v
[6.2 Numerical error]
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v
[6.3 Accumulation and reconciliation]
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v
[6.4 Series for forecasting]
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v
[Worked example] --> [Exercise EX016]
Learning outcomes
You will (1) use integrals to compute volumes from flow rates; (2) estimate numerical errors; (3) relate accumulation to reconciliation.
6.1 From flow to volume
Tidewater's sensors record flow rates every fifteen minutes. Volume supplied over a day is the integral of flow over time. Kofi approximates it with the trapezoidal rule on synthetic data and compares it with the volume calculated from the district's bulk meter. The two agree within half a percent. He explains that this calculation feeds the leakage estimate, so the method used to integrate flow is itself a data processing decision that should be documented and applied consistently. The method is written into the reporting standard.
6.2 Numerical error
Kofi tests how sampling interval affects error. With hourly samples instead of fifteen-minute samples, the trapezoidal estimate misses short peaks in demand and differs from the bulk meter by two percent. He shows that halving the interval roughly quarters the error for smooth flows, as theory predicts, but not for sudden spikes. The finding informs a governance question: what sampling interval is required for the data to be fit for leakage reporting, and who sets that requirement? Leakage reporting needs fifteen-minute data.
6.3 Accumulation and reconciliation
Integrals accumulate. Small errors in flow add up over a month into large differences in volume. Kofi computes a monthly reconciliation between supplied volume from sensors and billed volume from meters. The difference represents leakage plus measurement error plus timing differences in meter reading. He learns that reconciliation is a central data governance control in utilities, because it connects two independent data sources and reveals when either has a problem. Each month's difference is tracked over time, and a sudden jump prompts a check of both feeds before anyone concludes that leakage has increased.
6.4 Series for forecasting
Kofi uses a simple series expansion to approximate a demand curve that rises in the morning. The first terms capture most of the shape, while later terms add little. Prof. Varga explains that the same idea appears in data governance: a small number of well-chosen quality measures often captures most of the risk, while many extra measures add cost and confusion. Kofi records this as a design principle for the measurement work he will do in later chapters. Fewer, better measures win.
Worked example
Kofi integrates flow with the trapezoidal rule within half a percent of the bulk meter, shows hourly sampling raises error to two percent and links accumulation to monthly reconciliation.
Synthetic case. Flow data are synthetic.
Errors and limits
The trapezoidal rule assumes smooth flow between samples; spikes break it.
Exercises
- EX016 — Integration agreement. How closely did the trapezoidal estimate match the bulk meter? Rubric: 3 = within half a percent;; 2 = partly correct;; 1 = wrong;; 0 = none.
- EX017 — Hourly error. What error did hourly sampling produce? Rubric: 3 = about two percent;; 2 = partly correct;; 1 = wrong;; 0 = none.
- EX018 — Reconciliation components. What does the supplied-minus-billed difference contain? Rubric: 3 = leakage, measurement error, timing differences;; 2 = all;; 1 = one;; 0 = none.
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