Kap 3 wrap — techniques & solids sketch

V1 — Light substitution: rename the pile.

Builds on: compute / FTC Tool + chain idea from derivatives — undo a chain inside an integrand.

Tool gained: light u-sub — if integrand looks like f(g(x)) g′(x), set u=g(x), du=g′(x)dx, integrate in u, return.

1. Principle. A definite pile can be renamed when the inside function’s derivative sits next to it — same FTC, cleaner variable.
2. Model. ∫ f(g(x)) g′(x) dx = ∫ f(u) du with u=g(x).
3. Prediction / Answer. ∫_0^1 2x cos(x²) dx: u=x², du=2x dx → ∫_0^1 cos u du = sin1 − sin0 ≈ 0.841. Transfer: chain undo, not a new theory.

V2 — Volume of revolution (one vortex max).

Builds on: area-under Tool + definite integral (apps/area). Span +1: stack disks whose area is πr². Only this one solids Vortex — no shells/washers course leap.

Tool gained: disk sketch — rotate y=f(x) about the x-axis: V ≈ ∫_a^b π [f(x)]² dx.

1. Principle. Slice the solid into thin disks; each disk volume is πr² Δx with r=f(x); pile them with a definite integral.
2. Model. V = ∫_a^b π [f(x)]² dx (rotation about x-axis; sketch only).
3. Prediction / Answer. Rotate y=√x on [0,4] about x-axis: V=∫_0^4 π x dx = 8π. Transfer: a turned bowl’s capacity from a profile curve — one solids Tool, stop.

V3 — Mixed transfer back to apps Tools.

Builds on: V1 u-sub + V2 disk + apps (net change / average). Span stays honest: reuse, don’t open probability/solids encyclopedias.

Tool gained: Kap 3 transfer pack — pick sub, disk, or net-change/average for the job in front of you.

1. Principle. Wrap means matching the job to the Tool: rename pile, spin a profile, or integrate a rate — not every technique in the book.
2. Model. Job → choose ∫f, u-sub, π∫f², or (1/(b−a))∫f.
3. Prediction / Answer. Briefing: average heat from apps + a turned clay volume from disk — two Tools, one shop order. Heavier Kap 3 leftovers stay named Soon only if truly beyond span.