Math / Helper companion / Integral intro
Uncut · book companion · ~150–158
Pile-up & undo the slope tool
Area & antiderivative
Derivative asked “how steep?” Integrals ask “how much piles up?” Climb: stack thin strips under a graph → name a function whose derivative recovers the height (antiderivative) → indefinite integral +C. This rung must use the derivative Tool as reverse — not a brand-new symbol dump. (integral · primitiv funktion · area)
V1 — Area as stacked thin strips.
Builds on: reading f(x) as height on a graph (and slope sense from derivative intro).
Tool gained: approximate area under f on [a,b] ≈ Σ f(xᵢ*) Δx (rectangle / Riemann strip sum).
- Principle. Under a curve, thin vertical strips look like rectangles: height ≈ f at a sample, width Δx; stack them to estimate how much piles up.
- Model. Area ≈ Σ f(xᵢ*) Δx; finer Δx → better estimate (toward the definite integral later).
- Prediction / Answer. Under f(x)=x on [0,2]: two strips of width 1 with right heights 1 and 2 → ≈ 1+2 = 3; true triangle area is 2 — strips improve as you refine.
V2 — Antiderivative: undo the derivative.
Builds on: f′ as a function / power shortcuts (derivative Tool) — reverse the slope map.
Tool gained: antiderivative F of f means F′ = f (on an interval).
- Principle. If differentiation maps F → f, asking “who has slope f?” is the reverse job — same Tool run backward.
- Model. F is an antiderivative of f when F′(x) = f(x). Example: F(x)=x³/3 undoes f(x)=x² because (x³/3)′ = x².
- Prediction / Answer. f(x)=3x² → one antiderivative is x³ (check: (x³)′=3x²). Transfer: if speed is v(t), position is an antiderivative of v.
V3 — Family +C; indefinite integral mark.
Builds on: antiderivative F (V2) — constants share the same derivative.
Tool gained: indefinite integral ∫ f(x) dx = F(x) + C when F′ = f.
- Principle. Adding a constant does not change slope, so every antiderivative is F+C — the +C is not decoration; it is the whole family.
- Model. ∫ f(x) dx = F(x) + C. Differentiating both sides recovers f.
- Prediction / Answer. ∫ 2x dx = x² + C. Check: (x²+C)′ = 2x. Next companion spends this undo tool with endpoints (FTC).
1. Poor dump → rewrite
Book dump: “∫ means antiderivative” with a box of symbols and no reverse-of-derivative story. Our rewrite: strips for pile-up first, then undo the slope Tool you already earned, then name +C as the family.
Weapons: strip sum → antiderivative F′=f → ∫f = F+C. Next: compute with power reverse and FTC (endpoints).
2. Hands-on — strips under y = x
Approximate ∫₀² x dx with n equal right rectangles. True area = 2.
…
3. Fake → real
Fake
“Find ∫ 3x² dx.” Symbol in, +C out — no job.
Real
A tank fills at rate r(t)=3t² L/min. An antiderivative of r is t³ (+C). From empty at t=0, volume at t=2 is 8 L — same undo of derivative, now a fill job.
Gate — four, all required
1. Reconstruct: an antiderivative F of f means…
2. Why does V2 build on the derivative Tool (not brand-new magic)?
3. Predict: ∫ 4x³ dx = ?
4. Transfer: speed v(t) known; what is position?