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Engineering · Chapter 6 · English

Chapter 6 — Calculus II

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Chapter 6 — Calculus II

Lesson map

Lesson 06 published. Map, example, EX016.

flowchart TD
  N0["Learning outcomes"]
  N1["6.1 Totals from rates"]
  N2["6.2 Exponential decay and growth"]
  N3["6.3 Retention estimates"]
  N4["6.4 Checking with data"]
  N5["Growth and headroom"]
  N6["Synthesis"]
  N0 --> N1 --> N2 --> N3 --> N4 --> N5 --> N6
  N6 --> EX["Worked example"]
  EX --> Q["Exercise EX016"]

Logical drawing

flowchart LR
  N0["Learning outcomes"]
  N1["6.1 Totals from rates"]
  N2["6.2 Exponential decay and growth"]
  N3["6.3 Retention estimates"]
  N4["6.4 Checking with data"]
  N5["Growth and headroom"]
  N6["Synthesis"]
  N0 --> N1 --> N2 --> N3 --> N4 --> N5 --> N6 --> Q["EX016"]

[Learning outcomes]
    |
    v
[6.1 Totals from rates]
    |
    v
[6.2 Exponential decay and growth]
    |
    v
[6.3 Retention estimates]
    |
    v
[6.4 Checking with data]
    |
    v
[Growth and headroom]
    |
    v
[Synthesis]
    |
    v
[Worked example] --> [Exercise EX016]

Learning outcomes

You will (1) use integrals to compute totals from rates; (2) apply exponential models; (3) estimate data volumes.

6.1 Totals from rates

If throughput r(t) is known over a day, the integral gives total data transferred. Kofi integrates a fitted rate curve for the nightly backup to estimate daily volume over the inter-office link, which feeds capacity planning for the new encrypted tunnels.

If a link carries data at a known rate, the total transferred is the rate multiplied by time, or the area under the rate curve when the rate varies. Engineers use this to estimate log volumes and backup sizes. For example, a sensor producing two megabytes of logs per minute generates about 2.9 gigabytes per day, which tells Kofi how much storage a retention period needs before any purchase.

6.2 Exponential decay and growth

Some quantities change in proportion to their size. The probability that a link has not yet failed often follows an exponential model, and log volumes may grow roughly exponentially as more devices are added. Kofi uses these models carefully as approximations.

The same proportional rule appears in capacity planning, where Kofi has to decide whether a trend is a straight line or a curve. Traffic that grows by a fixed percentage each month grows exponentially, so capacity that seems ample today can run out sooner than a straight-line estimate suggests. Decay works the other way: the share of devices still running an old firmware version often halves over a regular period. Fitting the right model prevents both overspending and surprise shortfalls.

6.3 Retention estimates

Logging every allowed and denied flow creates large volumes. If flow records arrive at a rate that grows over the day, integrating gives daily volume, and multiplying by retention days gives storage needs. This helps size log storage for the ninety days the council requires.

Retention planning combines daily volume, compression, and the number of days records must be kept. Thirty days of three gigabytes per day is ninety gigabytes before compression, and engineers add headroom for growth and incident surges, when logging often increases sharply. The required period comes from policy and investigation needs, not from what happens to fit on an existing disk.

6.4 Checking with data

Models are checked against measured totals. If an integral predicts 50 GB but measurements show 70 GB, the model is wrong and Kofi revises it.

Models are only useful if they match reality, so engineers compare estimates with measured values from a pilot. If measured log volume is double the estimate, the assumptions need revisiting before the full rollout.

Engineers record each assumption, such as events per device per minute and average record size, so a mismatch can be traced to the specific assumption that was wrong. The corrected model then becomes the basis for the full plan.

Growth and headroom

Capacity plans must look ahead. If the number of council devices grows by about eight percent a year, log volume grows roughly in proportion, and compound growth over five years raises volume by nearly half. Kofi models this as exponential growth and plans storage with headroom for five years, then checks the plan yearly against actual volume. He also notes that growth is not smooth: a new office or a camera upgrade can add a step change, so the yearly review asks the council about planned projects as well as measuring trends.

Worked example

With r(t) = 2 + sin(πt/12) GB per hour over 24 hours, the integral is 48 GB because the sine term integrates to zero over a full period.

Synthetic case. Rates are synthetic.

Errors and limits

Integrating a poor model gives precise-looking wrong answers; validate against measurement.

Exercises

Synthesis

Integration turns rates into volumes for capacity and storage planning.

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