This area of study extends the techniques of differentiation and integration to solve complex geometric and physical problems. It includes the calculation of arc lengths, surface areas, volumes of revolution, and the evaluation of improper integrals and differential equations.
The Fundamental Theorem of Calculus connects differentiation and integration, allowing the calculation of definite integrals using antiderivatives.
If $f$ is a continuous function on the interval $[a, b]$, then:
$$\int_{a}^{b} f(x) \, dx = [F(x)]_a^b = F(b) - F(a)$$
where $F’(x) = f(x)$.
EXAM TIP: When calculating the total area, always sketch the graph or find the $x$-intercepts first to determine where the function is negative. If you use a CAS, the absolute value function
abs()is your best friend.
Integration can be used to find the exact length of a smooth curve $y = f(x)$ between two points $x = a$ and $x = b$.
The arc length $s$ is given by:
$$s = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx$$
If the curve is defined parametrically by $x = x(t)$ and $y = y(t)$, the arc length from $t = t_1$ to $t = t_2$ is:
$$s = \int_{t_1}^{t_2} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt$$
STUDY HINT: Arc length integrands often involve square roots of polynomials. If solving by hand, look for perfect square trinomials under the radical that allow the square root to be simplified.
A solid of revolution is formed by rotating a region in the plane about an axis.
| Axis of Rotation | Formula |
|---|---|
| $x$-axis (from $x=a$ to $x=b$) | $V = \pi \int_{a}^{b} [f(x)]^2 \, dx$ |
| $y$-axis (from $y=c$ to $y=d$) | $V = \pi \int_{c}^{d} [g(y)]^2 \, dy$ |
When a curve $y = f(x)$ is rotated about the $x$-axis, the area $S$ of the resulting surface is:
$$S = 2\pi \int_{a}^{b} y \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx$$
COMMON MISTAKE: Students frequently forget to include the $\pi$ in the volume formula or the $2\pi$ in the surface area formula. Always double-check your constants before finalizing your answer.
An integral is “improper” if the interval of integration is infinite or if the integrand becomes infinite (discontinuous) within the interval.
$$\int_{a}^{\infty} f(x) \, dx = \lim_{k \to \infty} \int_{a}^{k} f(x) \, dx$$
If $f(x)$ is discontinuous at $x = b$:
$$\int_{a}^{b} f(x) \, dx = \lim_{k \to b^-} \int_{a}^{k} f(x) \, dx$$
KEY TAKEAWAY: You cannot simply “plug in” infinity. VCAA requires the use of limit notation to demonstrate a formal understanding of improper integrals.
A differential equation (DE) involves an unknown function and its derivatives. In Specialist Maths, we focus on First-Order Separable Variables.
If $\frac{dy}{dx} = g(x)h(y)$, then:
$$\int \frac{1}{h(y)} \, dy = \int g(x) \, dx$$
Calculus is used to model real-world rates of change:
1. Growth and Decay: $\frac{dy}{dt} = ky$ (Exponential) or $\frac{dy}{dt} = ky(L - y)$ (Logistic).
2. Newton’s Law of Cooling: $\frac{dT}{dt} = -k(T - T_s)$, where $T_s$ is the surrounding temperature.
3. Mixing Problems (Tanks):
$$\frac{dQ}{dt} = \text{Rate In} - \text{Rate Out}$$
$$\frac{dQ}{dt} = (c_{in} \cdot r_{in}) - \left(\frac{Q(t)}{V(t)} \cdot r_{out}\right)$$
VCAA FOCUS: Modelling questions often require you to “Set up a differential equation” from a word problem. Pay close attention to the units and the sign (positive for increasing, negative for decreasing) of the rates.
| Application | Integral Formula | Key Considerations |
|---|---|---|
| Area | $\int | f(x) - g(x) |
| Arc Length | $\int \sqrt{1 + (y’)^2} dx$ | Derivative must be squared. |
| Volume ($x$-axis) | $\pi \int y^2 dx$ | Square the function, not the integral. |
| Surface Area ($x$-axis) | $2\pi \int y \sqrt{1 + (y’)^2} dx$ | Combination of radius and arc length. |
| Euler’s Method | $y_{new} = y_{old} + h \cdot \frac{dy}{dx}$ | Numerical approximation; smaller $h$ is more accurate. |
REMEMBER: For arc length and surface area, the term $(y’)^2$ refers to $\left(\frac{dy}{dx}\right)^2$. Forgetting to square the derivative is a frequent source of error in high-pressure exam conditions.
