Integration and Area Under a Curve: 0606 Exam Technique

Integration in 0606 is differentiation’s mirror, and in exam terms it is slightly easier to score on, because the question patterns are more predictable. Here is the method, in mark-scheme order.

The core skill: reverse the power rule

$\int x^n \,dx = \frac{x^{n+1}}{n + 1} + c$ ($n \ne -1$). Raise the power, divide by the new power, add c. The dropped “+c” is one of the most-reported errors in 0606 examiner comments, it costs the final accuracy mark on otherwise perfect working, which is the most painful kind of lost mark. Make it mechanical: finish every indefinite integral by writing $+ c$ before you even simplify.

The other standard forms you must know (none are given): $\int (ax + b)^n \,dx = \frac{(ax + b)^{n+1}}{a(n+1)} + c$, plus the integrals of $e^{ax+b}$, $\sin(ax + b)$ and $\cos(ax + b)$, each “divide by the inside coefficient”. They’re all on the formulae list.

Definite integrals: the bracket habit

For $\int_a^b$, the working examiners want to see is:

$[F(x)]$ with limits, $F(b) - F(a)$, value

Write the substituted line before computing. On the non-calculator paper, that visible substitution line is where the method mark sits, and the arithmetic is designed to fall out in exact fractions, a clean answer is a quiet confirmation you’re right.

Area under a curve: the four-step routine

1. Sketch. Thirty seconds, axes crossings marked. This single habit prevents the classic disaster: integrating across a region where the curve dips below the x-axis and “losing” area to sign cancellation.
2. Identify the region. Between curve and x-axis? Between curve and line? Between two curves?
3. Set up the integral(s). Below-axis portions: integrate separately and take the magnitude. Curve-and-line regions: $\int (\text{top} - \text{bottom}) \,dx$ between intersection points, find those by solving simultaneously.
4. Evaluate with visible substitution, exact values until the final line.

A frequent twist: the question gives a derivative result earlier (“show that $\frac{dy}{dx} = \ldots$”), then says “hence” find the integral, connected-parts questions where the command word is telling you the route. If your integral doesn’t use the earlier part, re-read the question.

Kinematics by integration

Velocity integrates to displacement; acceleration integrates to velocity, with the constant found from initial conditions (“initially at rest” means $v(0) = 0$, and that statement earns its own mark when written down). The full motion framework is in the calculus topic notes.

Practice that converts to marks

Integration rewards pattern drilling more than any other 0606 skill: do ten mixed integrals daily for a week, then a past-paper area question daily for another week, marking with the real scheme. Track dropped marks by cause, the fixes for “forgot +c”, “no sketch” and “sign error in $F(b) - F(a)$” are all habits, not knowledge.

In our online 1-to-1 classes (RM80/hr, 1.5 hours, anywhere in Malaysia) integration working is marked line by line against 0606 schemes until those habits are automatic. The first hour is a free trial, book it on WhatsApp.