Integration is the reverse of differentiation, and every skill in this chapter grows out of that one idea. This page shows how the skills connect, the order to study them in and where to start.
How do the skills connect?
Start with a derivative. If dy/dx = 3x², then y = x³ + c, because the derivative of x³ + 5 and of x³ − 2 is the same. The constant c is what a gradient cannot tell you.
A second piece of information fixes it. If the curve passes through (1, 4), then 1 + c = 4, so c = 3 and y = x³ + 3.
The remaining skills reuse this. A definite integral puts limits in place of the constant, and what you calculate from it depends on the question: an area, a volume or a distance.
What order should I study them in?
- Finding an indefinite integral and its constant builds the basic rule.
- Evaluating definite integrals adds limits.
- Finding area between a curve and an axis gives the integral a picture.
- Finding area between two curves extends it.
- Calculating volumes of revolution squares the function first.
- Connecting integration to displacement and distance links it to motion.
Finish with the chapter practice set.
Who should start where?
A student who is unsure how to integrate x⁻² or an expression in brackets should start at step 1. A student who integrates correctly but loses marks on signs and limits should start at step 2.
A student who finds areas but gets zero or negative answers should go to step 3, and a student who has met motion problems in another chapter may prefer step 6 first, together with kinematics of linear motion.
To have a teacher watch your working, see online one-to-one Additional Mathematics tuition. The wider SPM Additional Mathematics guide shows how integration fits with differentiation.