Integration in SPM Additional Mathematics: Antiderivatives, Areas, and Volumes
Integration is differentiation in reverse, and SPM treats it that way: if you understand differentiation, integration is mostly a matter of direction. This guide covers the standard integrals, the constant of integration, definite integrals, and the two geometric applications examiners love — areas under curves and volumes of revolution.
The Reverse Power Rule
If differentiating axⁿ gives a·n·xⁿ⁻¹, then integrating axⁿ gives a·xⁿ⁺¹ / (n + 1) plus a constant C, as long as n ≠ -1. Always add the constant for indefinite integrals; for definite integrals between limits, the constant cancels out.
| Function | Integral | Notes |
|---|---|---|
| axⁿ (n ≠ -1) | a·xⁿ⁺¹ / (n+1) + C | Reverse power rule |
| sin x | -cos x + C | Note the minus |
| cos x | sin x + C | |
| sec² x | tan x + C | |
| eˣ | eˣ + C | |
| 1/x | ln|x| + C | Only valid for x > 0 at SPM |
Definite Integrals
A definite integral has upper and lower limits, written as the integral from a to b of f(x) dx. To evaluate, find the antiderivative, substitute the upper limit, then subtract the value at the lower limit. The constant of integration is not needed because it cancels. Definite integrals give a number, not a function.
Finding Area Under a Curve
The area between the curve y = f(x), the x-axis, and the vertical lines x = a and x = b is the definite integral of f(x) from a to b — but only when the curve lies above the x-axis. If the curve dips below the axis, the integral gives a negative value, and you must take the absolute value or split the integral at the x-intercept and add the magnitudes.
For the area between two curves, integrate the upper curve minus the lower curve between the points of intersection. Always sketch first so you know which curve is on top; integrating the wrong way gives a negative area and loses marks.
Volume of Revolution
When a region bounded by y = f(x) and the x-axis between x = a and x = b is rotated fully about the x-axis, the volume swept out is π times the integral of [f(x)]² from a to b. Square the function before integrating — a very common mistake is to integrate the function first and then square the result. Always write the squared function explicitly: π ∫ y² dx.
Integration as Reverse of Differentiation
Many SPM questions give dy/dx and ask for y, or give the gradient function plus a point and ask for the equation of the curve. Integrate to find y in terms of x, then substitute the given point to find the constant C. Without the point, you cannot determine C, so always check the question for boundary information.
Common Mistakes
- Forgetting to add C for indefinite integrals.
- Forgetting the minus sign when integrating sin x.
- Squaring after integrating instead of before, in volume questions.
- Not sketching the curve, leading to wrong area signs.
- Confusing the limits order in definite integrals (upper minus lower).
How to Revise
Pair every differentiation practice session with an integration one. The two are inseparable, and fluency in one builds the other. Past-year SPM Paper 2 questions on areas and volumes are predictable in structure, so once you can solve five in a row correctly you have effectively mastered the topic. Show all substitution steps; examiners award method marks even when the final number slips.