This module uses integration to answer two kinds of question: how much area lies between a curve and an axis, and how far and where a particle has moved. One idea runs through both: a definite integral adds up signed pieces, so you must watch where the sign changes.
Check the current Cambridge Additional Mathematics 0606 syllabus for the exact content and calculator rules in your exam year. Our Additional Mathematics learning guide shows where this module sits among the others.
What should you already know?
You need to integrate powers and evaluate a definite integral, which is covered in integration methods. For the motion lessons, you also need differentiation and rates from tangents, normals and rates. Factorising quadratics quickly will save time when you look for roots.
An orienting example
A particle moves with velocity v = t² − 4 m/s for 0 ≤ t ≤ 3. Find the displacement and the distance travelled.
Step 1, find when v = 0: t² − 4 = 0, so t = 2 (t = −2 is outside the interval).
Step 2, integrate: s = t³/3 − 4t.
Step 3, values: s(0) = 0, s(2) = 8/3 − 8 = −16/3, s(3) = 9 − 12 = −3.
Step 4, displacement: s(3) − s(0) = −3 m.
Step 5, distance: from 0 to 2 the particle moves 16/3 m backward. From 2 to 3 it moves −3 − (−16/3) = 7/3 m forward. Total = 16/3 + 7/3 = 23/3 m.
That one question used a root, an integral and a sign check. The same steps find the area between a curve and the x-axis, because velocity plays the role of y.
In which order should you study it?
- Find area between a curve and an axis: the base idea, F(upper) − F(lower), for a curve that stays above the axis.
- Split an interval when signed areas change: why the integral can cancel, and how to split at roots to get the true area.
- Calculate displacement from velocity: the same integral, now read as a change in position.
- Distinguish distance travelled from displacement: uses the splitting idea to tell the two apart when a particle turns round.
- Connect position, velocity and acceleration: the full chain, differentiating one way and integrating the other, with constants fixed by conditions.
Then work through the mixed practice set. One lesson a day and the practice set at the weekend is a steady pace.
Which traps catch most students here?
- Substituting only the upper limit and forgetting to subtract the lower one.
- Integrating across a root, so that positive and negative parts cancel and the area looks too small.
- Giving the displacement when the question asks for the distance.
- Using s = vt when velocity is a formula that changes with t.
- Missing the constant when integrating to find velocity or position.
- Using constant-acceleration formulas when acceleration contains t.
Each lesson shows one of these slips in full and then corrects it.
How should you use the practice set?
Do the twelve questions in order, on paper, with limits and signs written out. Afterwards, use the routing list at the end of the practice set to send each error to the right lesson.
Keep a short record of the error types. The mistake log and retest queue can store them so you can retest a week later.
Students who follow each lesson but freeze on a mixed question often need someone to watch how they start. That is the kind of work we do in online one-to-one Additional Mathematics tuition.