Overview
This chapter uses the definite integral to compute areas under and between curves and volumes of solids of revolution.
1. Properties of Definite Integrals
∫ₐᵇ f(x)dx = −∫ᵇₐ f(x)dx
∫ₐᵇ f(x)dx = ∫ₐᶜ f(x)dx + ∫꜀ᵇ f(x)dx
∫ₐᵇ f(x)dx = ∫ₐᵇ f(a+b−x)dx
∫₋ₐᵃ f(x)dx = 2∫₀ᵃ f(x)dx (even) ; = 0 (odd)
2. Area
Area under curve: A = ∫ₐᵇ y dx
Between two curves: A = ∫ₐᵇ (y_top − y_bottom) dx
3. Volume of Revolution
About x-axis: V = π ∫ₐᵇ y² dx
About y-axis: V = π ∫꜀ᵈ x² dy
Worked Examples
Example 1 — Area under a curve
Area under y = x² from x = 0 to x = 3.
A = ∫₀³ x² dx = [x³/3]₀³ = 27/3 = 9 sq.units
Example 2 — Using an even-function property
Evaluate ∫₋₂² x² dx.
x² is even → = 2∫₀² x² dx = 2[x³/3]₀² = 2(8/3) = 16/3
Example 3 — Area between curves
Area between y = x and y = x² from x = 0 to 1.
A = ∫₀¹ (x − x²) dx = [x²/2 − x³/3]₀¹ = 1/2 − 1/3 = 1/6 sq.units
Example 4 — Volume of revolution
Volume when y = x is rotated about the x-axis from 0 to 1.
V = π∫₀¹ x² dx = π[x³/3]₀¹ = π/3 cubic units
Common Mistakes
- Ignoring the modulus for areas below the x-axis.
- Subtracting in the wrong order (top − bottom) for area between curves.
- Forgetting the π in the volume formula.
- Mismatched limits when changing from dx to dy.
Exam Tips & Revision Checklist
- Use even/odd properties to shortcut symmetric integrals.
- Sketch the region first for area-between-curves problems.
- Practised: area under curve, even-function property, area between curves, volume of revolution.
Frequently Asked Questions
Area = ∫ₐᵇ y dx for a curve y = f(x) between x = a and x = b (take the modulus for parts below the x-axis).