Math / Helper companion / Integral area
Uncut · book companion · ~159–167
Spend the definite Tool on shapes
Area under & between curves
FTC gives numbers from antiderivatives. Climb: area under one nonnegative graph → area between two graphs as ∫(upper−lower) → signed area vs geometric area when f dips below. Uses the compute Tool — not a new chapter of unrelated geometry. (area · mellan kurvor · tecken)
V1 — Area under one curve.
Builds on: ∫_a^b f = F(b)−F(a) (compute / FTC Tool).
Tool gained: for f ≥ 0 on [a,b], geometric area under y=f equals ∫_a^b f(x) dx.
- Principle. The definite integral that undoes slope also counts the stacked strips under a nonnegative graph — same number, area job.
- Model. Area = ∫_a^b f when f≥0 on the interval.
- Prediction / Answer. Under y=4−x² from −2 to 2: ∫(4−x²)=[4x−x³/3]₋₂² = 32/3. Transfer: paint needed under a parabolic arch model.
V2 — Area between two curves.
Builds on: area-under Tool (V1) — difference of heights is still an integrand.
Tool gained: area between y=f and y=g on [a,b] = ∫_a^b |f−g| dx (or ∫(upper−lower) when upper is clear).
- Principle. Between two graphs, each strip’s height is top minus bottom — one integral of the gap, not a brand-new topic.
- Model. If f≥g on [a,b], Area = ∫_a^b (f−g) dx.
- Prediction / Answer. Between y=x and y=x² on [0,1]: ∫(x−x²)=[x²/2−x³/3]_0^1 = 1/6.
V3 — Signed vs geometric area.
Builds on: between-curves / under-curve Tools — split where f changes sign.
Tool gained: ∫_a^b f is net signed area; geometric area uses ∫|f| or split at zeros.
- Principle. Below the axis, strips subtract in the plain definite integral. A paint job that wants positive area must take absolute values or split intervals.
- Model. Net = ∫ f; geometric = Σ ∫|f| on pieces where sign is constant.
- Prediction / Answer. ∫₋₁¹ x dx = 0 (net), but geometric area under |x| on [−1,1] is 1. Next: light quantity apps with the same definite Tool.
1. Poor dump → rewrite
Book dump: “area = ∫” and “between = ∫(f−g)” as two chants. Our rewrite: each spends the definite Tool — one curve, then gap height, then sign care.
Weapons: area under f≥0 → ∫(upper−lower) → signed vs |f|. Next: net change / average / light quantity.
2. Hands-on — gap between x and x²
On [0,1], upper−lower = x−x². Drag the right endpoint c ∈ (0,1].
…
3. Fake → real
Fake
“Find the area between y=x and y=x² on [0,1].” No reason.
Real
A solar panel’s clear aperture is the region between a linear frame edge and a curved shade model on a unit map. Area 1/6 (map units) sizes the glass order — same ∫(upper−lower), now a materials job.
Gate — four, all required
1. Reconstruct: area under f≥0 on [a,b] is…
2. Why is between-curves a climb from under-curve (not brand-new)?
3. Predict: area between y=x and y=x² on [0,1]?
4. Transfer: ∫₋₁¹ x dx = 0. Geometric area of |x|?