Math / Helper companion / Integral compute
Uncut · book companion · ~154–162
Spend the undo tool — then endpoints
Computing integrals & FTC
You have antiderivatives and +C. Climb: reverse the power/sum shortcuts → evaluate F(b)−F(a) for a definite pile-up → name that bridge the Fundamental Theorem. Must use the intro Tool; not a leap to every technique at once. (bestämda integraler · primitiv · analysens fundamentalsats)
V1 — Reverse power / sum shortcuts.
Builds on: ∫f = F+C and (xⁿ)′ = n xⁿ⁻¹ (intro + derivative shortcuts).
Tool gained: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n≠−1); linearity of ∫.
- Principle. Differentiation’s power rule run backward raises the exponent and divides — same family of shortcuts you already trust going forward.
- Model. For n≠−1, ∫ xⁿ dx = xⁿ⁺¹/(n+1)+C; ∫(cf+g)=c∫f+∫g.
- Prediction / Answer. ∫(3x²−4) dx = x³ − 4x + C. Check by differentiating.
V2 — Definite: F(b) − F(a).
Builds on: antiderivative F from V1 / intro — pin C with two endpoints.
Tool gained: definite integral ∫_a^b f(x) dx = F(b) − F(a) when F′=f (continuous case).
- Principle. Net pile-up from a to b is the change in an antiderivative — C cancels, so you only need any one F.
- Model. ∫_a^b f = [F(x)]_a^b = F(b)−F(a).
- Prediction / Answer. ∫_0^2 3x² dx = [x³]_0^2 = 8. Same as net volume if fill rate is 3t² from 0 to 2.
V3 — FTC: strips ↔ antiderivative bridge.
Builds on: strip-sum idea (intro V1) + definite F(b)−F(a) (V2).
Tool gained: FTC reading — d/dx ∫_a^x f = f(x); and ∫_a^b f = F(b)−F(a) links area/net change to undo-slope.
- Principle. Accumulating area from a fixed start up to x, then differentiating in x, recovers the height f(x) — strip pile-up and antiderivative are two faces of one Tool.
- Model. If A(x)=∫_a^x f(t) dt then A′(x)=f(x). Combined with V2: compute area/net change via any antiderivative.
- Prediction / Answer. A(x)=∫_0^x (2t) dt = x² → A′(x)=2x, matching the integrand. Next: spend the definite Tool on geometric area jobs.
1. Poor dump → rewrite
Book dump: “FTC: ∫_a^b f = F(b)−F(a)” as a stamp with no climb from undo-slope. Our rewrite: reverse shortcuts → pin endpoints → name the bridge to strips.
Weapons: reverse power → F(b)−F(a) → FTC (A′=f and definite via F). Next: area under/between curves.
2. Hands-on — F(b)−F(a) live
f(x)=3x². Drag b; watch F(b)−F(0) with F(x)=x³.
…
3. Fake → real
Fake
“Evaluate ∫_1^4 2x dx.” Endpoints with no story.
Real
Current into a battery is i(t)=2t A. Charge from t=1 to 4 s is ∫_1^4 2t dt = [t²]_1^4 = 15 C — same F(b)−F(a), now a coulomb budget.
Gate — four, all required
1. Reconstruct: ∫_a^b f(x) dx equals…
2. Why does V2 need the antiderivative Tool from intro?
3. Predict: ∫_0^3 x² dx = ?
4. Transfer: A(x)=∫_2^x f. What is A′(x)?