Exercise Solutions Available
Step-by-step answers for all textbook questions in this chapter
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).