Revise Integration for Pure Mathematics 2 WMA12 (AS Level) — revision notes and instant AI marking.
📖 Revision notes · preview
The Big Idea
Integration is the reverse of differentiation. It lets you find the area under a curve by adding up infinitely many infinitely thin slices. Instead of memorising the formula, understand that ∫ is just a fancy way of saying "find the total by adding everything up".
Quick Summary: What You Need to Know
1. Definite Integration & The Fundamental Theorem
What is Definite Integration?
Definite integration means evaluating an integral between two specific limits (numbers). You find the antiderivative, then substitute the limits and subtract.
It's called "definite" because you get one definite number as your answer, not a family of functions.
The Fundamental Theorem of Calculus
∫ₐᵇ f'(x) dx = f(b) − f(a)
Why Is There No Constant of Integration?
When you integrate indefinitely, you add +c. But in definite integration, the constant cancels out.
f(b) = ... + c
f(a) = ... + c
f(b) − f(a) = (... + c) − (... + c) = ... [the +c's cancel!]
This is why you can ignore +c when doing definite integration — it will always vanish when you subtract anyway.
Step-by-Step: How to Calculate a Definite Integral
Example: Find ∫₂⁵ 9x² dx
Step 1 Integrate without applying limits yet
∫ 9x² dx = 9 · x³/3 = 3x³ (ignore +c)
Step 2 Use square bracket notation [ ]ₐᵇ to show the limits
[3x³]₂⁵
Step 3 Substitute the upper limit, then subtract the lower limit
= 3(5)³ − 3(2)³
= 3(125) − 3(8)
= 375 − 24
= 351
💡 Key Insight: The order matters! You always do [upper] − [lower], not the other way around. If you do it backwards, your answer will be negative.
Using a Calculator
Your scientific calculator can evaluate definite integrals directly (usually a button that looks like ∫ or has it in a menu). This saves time, but only use it if the question allows it — some exams require you to show the working by hand.
Practice Question 1
Find the exact value of ∫₁⁴ (3x² − 6x) dx
2. Area Under a Curve
The Basic Idea
The area between a curve, the x-axis, and two vertical lines is found by integrating the function.
Area Under a Curve
Area = ∫ₐᵇ y dx
Think of it this way: imagine the area under the curve as infinite vertical strips, each with width dx (infinitely small) and height y. When you integrate, you're adding up all those strips.
Area = sum of all infinitely thin rectangles
= ∫ₐᵇ (height) × (width) dx
= ∫ₐᵇ y dx