Over fourteen lessons, we have built a complete toolkit for understanding change and accumulation. We began with antiderivatives — the reverse of differentiation — and discovered the Fundamental Theorem of Calculus, which unifies differentiation and integration into a single coherent theory. We learned to calculate areas between curves and volumes of revolution, transforming geometric problems into algebraic ones. We mastered three integration techniques — substitution, by parts, and partial fractions — each unlocking a new class of solvable problems. Finally, we applied everything to differential equations, the language of dynamic systems. This lesson draws the threads together, showing how each piece connects to the others and preparing you for the HSC examination.
Use the PDF for classwork, homework or revision.
Core Content
Foundation: Antiderivatives and indefinite integrals (L01–L03)
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Core Theorem: The Fundamental Theorem of Calculus (L04–L05)
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Applications: Areas and volumes (L06–L07)
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Techniques: Substitution, by parts, partial fractions (L08–L10)
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Dynamic Systems: Differential equations (L11–L13)
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Physical Applications: Motion (L14)
Key connections:
| Error | Why it happens | How to avoid |
|---|---|---|
| Forgetting +C | Rushing through indefinite integrals | Always write +C; check if definite |
| Wrong power rule at n = -1 | Applying $\frac{x^0}{0}$ | Remember: $\int \frac{1}{x}\,dx = \ln|x| + C$ |
| Area below x-axis negative | Not taking absolute value | Sketch first; use |f(x)| for area |
| Washer: $(R-r)^2$ | Confusing with algebraic expansion | Square first, then subtract: $R^2 - r^2$ |
| Substitution: forgetting du | Not converting dx to du | Always write $\frac{du}{dx}$ and solve for dx |
| By parts: wrong u choice | Not using LIATE | LIATE: Log, Inverse, Algebraic, Trig, Exp |
| Distance vs displacement | Using v instead of |v| | Split integral where v changes sign |
5 random questions from a replayable lesson bank — feedback shown immediately
Short Answer
8. Evaluate $\int_0^{\pi/2} x \sin x \, dx$ using integration by parts. Show all working. 3 MARKS
9. Find the volume when $y = x^3$ from $x = 0$ to $x = 1$ is rotated around the x-axis. Then find the area between $y = x^3$ and $y = x$. Show all working. 4 MARKS
10. Write an essay-style response (150–200 words) explaining how the Fundamental Theorem of Calculus, integration techniques, and differential equations connect to form a unified theory of change. Use at least two real-world examples. 4 MARKS
1. $x^3 + \ln|x| + \frac{1}{2}e^{2x} + C$.
2. $u = x^2 + 1$, $du = 2x\,dx$. $\frac{1}{2}\int_1^5 \sqrt{u}\,du = \frac{1}{3}(5^{3/2} - 1) = \frac{1}{3}(5\sqrt{5} - 1)$.
3. $x^2 = 2 - x \Rightarrow x^2 + x - 2 = 0 \Rightarrow x = 1, -2$. $A = \int_{-2}^{1} (2 - x - x^2)\,dx = [2x - \frac{x^2}{2} - \frac{x^3}{3}]_{-2}^{1} = \frac{9}{2}$.
4. $\frac{dy}{y} = 3x^2\,dx$, $\ln|y| = x^3 + C$, $y = Ae^{x^3}$. $y(0) = 2$: $A = 2$. $y = 2e^{x^3}$.
5. $v(t) = t^2 + t + 2$, $x(t) = \frac{t^3}{3} + \frac{t^2}{2} + 2t + 1$. $x(3) = 9 + 4.5 + 6 + 1 = 20.5$ m.
1. FTC shows differentiation (instantaneous rate) and integration (accumulation) are inverse operations, connecting the two pillars of calculus.
2. Pharmacokinetic models may require integrating a product of polynomial and exponential (by parts) after decomposing a rational clearance function (partial fractions).
3. An antiderivative solves $\frac{dy}{dx} = f(x)$. A differential equation generalises this to $\frac{dy}{dx} = f(x,y)$, requiring more sophisticated techniques like separation of variables.
Q8 (3 marks): $u = x$, $dv = \sin x\,dx$, so $du = dx$, $v = -\cos x$ [1]. $[-x\cos x]_0^{\pi/2} + \int_0^{\pi/2} \cos x\,dx$ [1] $= 0 + [\sin x]_0^{\pi/2} = 1$ [1].
Q9 (4 marks): Volume: $V = \pi \int_0^1 x^6\,dx = \pi [\frac{x^7}{7}]_0^1 = \frac{\pi}{7}$ [2]. Area: $x^3 = x$ at $x = 0, 1$. On $[0,1]$, $x \geq x^3$. $A = \int_0^1 (x - x^3)\,dx = [\frac{x^2}{2} - \frac{x^4}{4}]_0^1 = \frac{1}{2} - \frac{1}{4} = \frac{1}{4}$ [2].
Q10 (4 marks): Response should mention: (i) FTC connects differentiation and integration as inverse operations [1]; (ii) integration techniques (substitution, by parts, partial fractions) extend the range of solvable problems [1]; (iii) differential equations use integration to find functions from rates of change [1]; (iv) at least two real-world examples with clear connection to the theory [1].
Climb platforms with mixed problems from across the entire Further Calculus module. Pool: lesson 15.
Tick when you've finished all activities and checked your answers.