Mathematics Advanced • Year 12 • Module 6 • Lesson 15
Module Synthesis — Further Calculus
Practise HSC-style writing across the whole module: by-parts, area/volume and a synthesis extended response.
1. Short-answer questions
1.1 Evaluate ∫₀π/2 x sin(x) dx using integration by parts. Show all working. 3 marks Band 5
1.2 Find the area between y = x³ and y = x on the interval [0, 1]. Show all working. 3 marks Band 4-5
1.3 Find the volume generated when y = x³ from x = 0 to x = 1 is rotated about the x-axis. Show all working. 2 marks Band 4
Stuck on 1.1? LIATE → u = x (Algebraic), dv = sin(x) dx (Trig).2. Extended response
2.1 Write a structured response (about 200 words plus a worked example) explaining how the Fundamental Theorem of Calculus, the standard integration techniques (substitution, by parts), and the methods for solving separable differential equations together form a unified toolkit for analysing change.
Your response must include:
(a) One clear statement of FTC Part 2 with brief explanation of what it lets you do.
(b) One short worked calculation using substitution OR by parts (one to three lines is enough).
(c) One short worked example of solving a separable DE (e.g. dy/dx = ky, y(0) = y0).
(d) Two real-world contexts (one for exponential growth/decay, one for motion/volume) showing where these tools are used.
(e) One closing sentence that explicitly names the unifying idea (e.g. "differentiation and integration are inverse operations"). 8 marks Band 5-6
Explicit marking criteria
Part (a) — 1 mark
• 1 mark — states FTC Part 2 (∫ab f(x) dx = F(b) − F(a)) and notes that it converts antiderivatives into signed areas.
Part (b) — 2 marks
• 1 mark — names the technique (substitution or by parts) and identifies u or u/dv correctly.
• 1 mark — reaches a correct final antiderivative or numerical value, with at least one line of working.
Part (c) — 2 marks
• 1 mark — separates correctly and integrates both sides.
• 1 mark — applies the initial condition to obtain a particular solution.
Part (d) — 2 marks
• 1 mark — one credible real-world example for growth/decay (e.g. carbon-14, COVID R0, continuous compounding).
• 1 mark — one credible real-world example for motion or volume (e.g. SpaceX guidance, braking distance, volume of a turbine blade).
Part (e) — 1 mark
• 1 mark — closes with an explicit unifying statement linking differentiation and integration as inverse operations (or equivalent).
Your response:
Stuck? Write each of (a)-(e) as its own short paragraph; markers reward clear, signposted structure.How did this worksheet feel?
What I'll revisit before next class:
1.1 — ∫₀π/2 x sin(x) dx (3 marks)
Sample response. u = x, dv = sin(x) dx ⇒ du = dx, v = −cos(x).
∫₀π/2 x sin(x) dx = [−x cos(x)]₀π/2 − ∫₀π/2 (−cos(x)) dx
= (0 − 0) + [sin(x)]₀π/2 = 1 − 0 = 1.
Marking notes. 1 mark — correct u, dv, du, v (LIATE applied). 1 mark — correct boundary term [−x cos(x)]₀π/2 = 0. 1 mark — correct final value 1. Common error: forgetting the minus sign on cos(x) gives the wrong final sign.
1.2 — Area between y = x³ and y = x on [0, 1] (3 marks)
Sample response. Intersections: x³ = x ⇒ x(x² − 1) = 0 ⇒ x = 0, x = 1 (also x = −1, outside our interval). On (0, 1), x > x³ (e.g. at x = 0.5: 0.5 > 0.125).
A = ∫₀¹ (x − x³) dx = [x²/2 − x⁴/4]₀¹ = 1/2 − 1/4 = 1/4.
Marking notes. 1 mark — correct intersections (0 and 1). 1 mark — identifies x as the top function on (0, 1) and sets up integrand correctly. 1 mark — correct evaluation. Students who write x³ − x lose the third mark (negative area).
1.3 — Volume of y = x³ rotated about x-axis on [0, 1] (2 marks)
Sample response. V = π ∫₀¹ (x³)² dx = π ∫₀¹ x⁶ dx = π [x⁷/7]₀¹ = π/7.
Marking notes. 1 mark — correct disk-method setup with [f(x)]² = x⁶. 1 mark — correct evaluation π/7. Common error: squaring afterwards (writing (x⁷/7)² instead of squaring the integrand first).
2.1 — Extended synthesis response (8 marks): sample Band-6 response with annotations
Sample Band-6 response.
(a) FTC Part 2. If F is an antiderivative of f on [a, b], then ∫ab f(x) dx = F(b) − F(a). This converts a question about area (the integral) into a question about antiderivatives (a single subtraction) — the central computational tool of integral calculus. [1 mark — statement + interpretation.]
(b) Worked example (substitution). Evaluate ∫ x ex² dx. Let u = x², so du = 2x dx and x dx = du/2. [1 mark — u choice and du.]
Then ∫ x ex² dx = (1/2) ∫ eu du = (1/2) eu + C = (1/2) ex² + C. [1 mark — correct antiderivative.]
(c) Worked example (separable DE). Solve dy/dx = ky with y(0) = y0. Separate: dy/y = k dx. Integrate: ln|y| = kx + C. [1 mark — separation and integration.]
Exponentiate: y = A ekx. Apply y(0) = y0: A = y0. So y(x) = y0 ekx. [1 mark — IC applied.]
(d) Real-world contexts.
• Growth/decay: Carbon-14 dating of fossils in the Naracoorte Caves uses dN/dt = −k N (k = (ln 2)/5730 per year); the same DE underpins continuous compounding of investments and the early-stage spread of an epidemic (COVID's R0). [1 mark.]
• Motion / volume: SpaceX Falcon-9 landing burns integrate a(t) twice per second to track position; the same calculus computes the volume of a turbine blade rotated about its axis using V = π ∫ [f(x)]² dx. [1 mark.]
(e) Unifying idea. Differentiation tells us the rate of change; integration tells us the total change. The Fundamental Theorem of Calculus makes precise the statement that these two operations are inverses, which is why every problem in Module 6 — whether about populations, particles, or pendulums — ultimately reduces to recognising a derivative and reversing it. [1 mark — explicit unifying statement.] ▮
Total: 8/8.
Band descriptors for marker.
Band 3: Mentions FTC vaguely; lists techniques without examples; only one real-world reference and no unifying statement. ≈ 2-3 marks.
Band 4: States FTC correctly; one short worked example correct; one real-world reference per area; no closing synthesis sentence. ≈ 4-5 marks.
Band 5: Both worked examples (substitution/by parts, separable DE) correct; two real-world contexts named but only one developed in detail; unifying statement present but generic. ≈ 5-6 marks.
Band 6: All five parts (a)-(e) addressed crisply, both worked examples mathematically sound, both contexts are specific and Australian-relevant where possible, closes with a sharp statement that names "inverse operations" or FTC explicitly. 7-8 marks.