Unit 4 extends the calculus toolkit with improper integrals, second-order ODEs, and optimisation/area/volume applications.
An improper integral has an infinite limit of integration or an unbounded integrand.
Type 1 (infinite limits):
$$\int_a^\infty f(x)\,dx = \lim_{b\to\infty}\int_a^b f(x)\,dx$$
Converges if the limit is finite; diverges otherwise.
Example: $\displaystyle\int_1^\infty \frac{1}{x^2}\,dx = \lim_{b\to\infty}\left[-\frac{1}{x}\right]1^b = \lim{b\to\infty}\left(-\frac{1}{b}+1\right) = 1$. Converges.
Compare: $\displaystyle\int_1^\infty \frac{1}{x}\,dx = \lim_{b\to\infty}[\ln x]_1^b = \infty$. Diverges.
Type 2 (discontinuous integrand):
$$\int_a^b f(x)\,dx = \lim_{c\to a^+}\int_c^b f(x)\,dx \quad \text{if } f \text{ is unbounded at } a$$
Example: $\displaystyle\int_0^1 \frac{1}{\sqrt{x}}\,dx = \lim_{c\to0^+}[2\sqrt{x}]_c^1 = 2$. Converges.
Area between $y = f(x)$ (upper) and $y = g(x)$ (lower) from $a$ to $b$:
$$A = \int_a^b [f(x) - g(x)]\,dx$$
Example: Area between $y = x$ and $y = x^2$ from $0$ to $1$:
$$A = \int_0^1 (x - x^2)\,dx = \left[\frac{x^2}{2} - \frac{x^3}{3}\right]_0^1 = \frac{1}{2}-\frac{1}{3} = \frac{1}{6}$$
Disk method (rotating $y = f(x)$ around the $x$-axis):
$$V = \pi\int_a^b [f(x)]^2\,dx$$
Washer method (rotating region between $y = f(x)$ and $y = g(x)$, $f > g$):
$$V = \pi\int_a^b \left([f(x)]^2 - [g(x)]^2\right)\,dx$$
Example: Volume from rotating $y = \sqrt{x}$ (from $x=0$ to $x=4$) around $x$-axis:
$$V = \pi\int_0^4 x\,dx = \pi\left[\frac{x^2}{2}\right]_0^4 = 8\pi$$
Find extreme values of $f(x)$ on a domain:
1. Find $f’(x)$ and solve $f’(x) = 0$ for critical points.
2. Test using second derivative $f’‘(x)$: $f’’ > 0$ is local minimum; $f’’ < 0$ is local maximum.
3. Compare with boundary values for absolute extrema.
KEY TAKEAWAY: Improper integrals require limit processes; always evaluate the limit explicitly. Area and volume calculations are powerful applications of the definite integral.
EXAM TIP: For volume of revolution, always square the radius function under the integral — the volume formula uses $[f(x)]^2$, not $f(x)^2$ halved.
VCAA FOCUS: Improper integrals (testing for convergence and evaluating) and volumes of revolution appear regularly in Paper 2 extended-response questions.
