Quantities & light applications

V1 — Net change from a rate.

Builds on: FTC / definite integral Tool (compute) — rate is the integrand.

Tool gained: net change of Q on [a,b] = ∫_a^b Q′(t) dt = Q(b)−Q(a).

1. Principle. If you know how fast something changes, integrating the rate recovers how much the quantity moved — same bridge as area, now a stock/flow job.
2. Model. ΔQ = ∫_a^b rate(t) dt.
3. Prediction / Answer. Water flows in at r(t)=3t² L/min from 0 to 2 → net +8 L. Transfer: charge from current, distance from speed.

V2 — Average value of f.

Builds on: definite integral of f (area/compute Tools) — divide by width.

Tool gained: average of f on [a,b] = (1/(b−a)) ∫_a^b f(x) dx.

1. Principle. The constant height whose rectangle matches the same area is the average value — strip pile-up divided by interval length.
2. Model. f_avg = (1/(b−a)) ∫_a^b f.
3. Prediction / Answer. f(x)=x² on [0,3]: ∫=9, width 3 → average 3. Transfer: average temperature from a sensor curve.

V3 — Light density sketch (area = 1).

Builds on: area under nonnegative f (area Tool) — normalize so total area is 1.

Tool gained: light continuous density reading — P(a≤X≤b) ≈ ∫_a^b p(x) dx when p≥0 and ∫_all p = 1 (sketch only).

1. Principle. When a nonnegative “height” curve is scaled so the whole area is 1, the area on a window is a share of the whole — probability-shaped, without a full stats leap.
2. Model. If ∫_{-∞}^{∞} p = 1 and p≥0, then ∫_a^b p is the mass on [a,b].
3. Prediction / Answer. Toy p(x)=1/2 on [0,2] (zero elsewhere): P(0.5≤X≤1.5)=∫_{0.5}^{1.5}(1/2)=1/2. Deeper climb continues in Kap 3 wrap (one solids vortex max).