Chapter 6 — Calculus II
Lesson map
Lesson 06 published. Map, example, EX016.
flowchart TD
N0["Learning outcomes"]
N1["6.1 Accumulation"]
N2["6.2 Manual clean-up"]
N3["6.3 Nightly restore"]
N4["6.4 Comparing designs"]
N5["6.5 Assumptions"]
N6["Checking with data"]
N7["Units"]
N0 --> N1 --> N2 --> N3 --> N4 --> N5 --> N6 --> N7
N7 --> EX["Worked example"]
EX --> Q["Exercise EX016"]
Logical drawing
flowchart LR
N0["Learning outcomes"]
N1["6.1 Accumulation"]
N2["6.2 Manual clean-up"]
N3["6.3 Nightly restore"]
N4["6.4 Comparing designs"]
N5["6.5 Assumptions"]
N6["Checking with data"]
N7["Units"]
N0 --> N1 --> N2 --> N3 --> N4 --> N5 --> N6 --> N7 --> Q["EX016"]
[Learning outcomes]
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v
[6.1 Accumulation]
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v
[6.2 Manual clean-up]
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v
[6.3 Nightly restore]
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v
[6.4 Comparing designs]
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v
[6.5 Assumptions]
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v
[Checking with data]
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v
[Units]
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v
[Worked example] --> [Exercise EX016]
Learning outcomes
You will (1) use accumulation and integrals to estimate total exposure; (2) compare designs by area under a curve.
6.1 Accumulation
If f(t) is the number of PCs holding leftover personal data at time t, the area under f over a month estimates total exposure in PC-days. Integrals measure accumulation.
A PC-day is one computer holding someone's data for one day. The measure combines how many PCs are affected and for how long, which is what matters for the patrons whose files are exposed. Two designs that leave the same number of PCs affected at a single moment can still differ greatly in total exposure if one clears them faster. Mateo chose this measure because it is easy to explain to managers and it captures both effects together.
6.2 Manual clean-up
With three new cases a day and fortnightly clean-ups, f rises from 0 to about 42 and drops: each fortnight's area is a triangle of about (14 × 42) / 2 = 294 PC-days.
The triangle comes from the straight-line rise between clean-ups and the drop to zero when staff clear the PCs. Mateo draws it on the whiteboard for Marta, because the shape makes the arithmetic obvious. A fortnight is Fernhill's actual cleaning interval at the time, taken from staff rotas rather than chosen for the calculation. The rota is attached to the calculation.
6.3 Nightly restore
With nightly resets, each day's triangle is (1 × 3) / 2 = 1.5 PC-days, or 21 per fortnight, far smaller.
The comparison uses the same arrival rate as the fortnightly case, so the only difference is how often the PCs are cleared. Each day starts clean, cases build up during opening hours and the overnight restore removes them. The small triangle is repeated fourteen times in a fortnight, which gives the total. Mateo checks the arithmetic in both cases with a short script that sums the daily counts, and the script agrees with the triangles. That agreement is the first check on the model.
6.4 Comparing designs
The ratio, about 14 to 1, quantifies the benefit. Engineers often compare designs by such areas: total exposure, total downtime or total cost.
Mateo is careful about what the ratio means. It compares two models under the same assumptions, so its accuracy depends on those assumptions holding for both. It does not say how many patrons were actually harmed. For a board decision it is enough: the nightly design reduces exposure by an order of magnitude, and the remaining exposure is limited to the hours of a single day.
6.5 Assumptions
The estimate assumes steady arrival of cases and full removal at each reset. Chapter 28 tests the second assumption.
If arrivals are not steady, for example during exam season, both designs see more exposure, but the ratio between them stays similar because the reset interval drives it. Full removal is the more fragile assumption. A restore that leaves some files behind would make the nightly design look better on paper than it is in practice, which is why it is tested directly rather than assumed. Mateo lists both assumptions in his report beside the result.
Checking with data
Mateo compares the integral estimate with Fernhill's records. Over two fortnights of manual clean-ups, staff found 39 and 45 PCs with leftover files, close to the model's peak of about 42. The match gives some confidence in the steady-arrival assumption for normal weeks. He notes that exam season brought more cases, so the model would underestimate exposure then.
The same reasoning, accumulating a quantity over time, appears later for downtime in chapter 32.
Units
He states units carefully: PC-days for exposure, PC-hours for downtime.
Beyond exposure
Integrals also estimate total effort. If staff spend about 20 minutes per manual clean-up of a PC, the fortnightly effort grows with the number of affected PCs. Nightly restore removes most of this effort, linking Calculus II to objective O4 in chapter 20.
Worked example
Dr Novak asks Mateo to state the result honestly. He writes: "Under steady-arrival assumptions, nightly restore reduces exposure from about 294 to about 21 PC-days per fortnight."
Synthetic case. Synthetic.
Errors and limits
Integrals of idealised functions give estimates, not measurements.
Exercises
- EX016 — Area meaning. What does the area under f represent? Rubric: 3 = total exposure in PC-days; 2 = partial; 1 = vague; 0 = none.
- EX017 — Fortnight estimates. Give the fortnight exposure for both designs. Rubric: 3 = 294 vs 21 PC-days; 2 = partial; 1 = vague; 0 = none.
- EX018 — Assumptions. Which assumptions underlie the estimate? Rubric: 3 = steady arrival, full removal; 2 = partial; 1 = vague; 0 = none.
Synthesis
Accumulation compares designs by total exposure.
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