Chapter 8 — Applications of Integrals — covers finding areas under curves and between curves using definite integrals. Carries 6 marks (typically one 6-mark question).
Key Concepts
Area Under a Curve
Area between curve y = f(x), x-axis, and x = a to x = b:
A = ∫ₐᵇ f(x) dx (when curve is above x-axis)
A = −∫ₐᵇ f(x) dx = |∫ₐᵇ f(x) dx| (when curve is below x-axis)
Area between curve x = f(y), y-axis, and y = c to y = d:
A = ∫ᶜᵈ f(y) dy
A = ∫ₐᵇ f(x) dx (when curve is above x-axis)
A = −∫ₐᵇ f(x) dx = |∫ₐᵇ f(x) dx| (when curve is below x-axis)
Area between curve x = f(y), y-axis, and y = c to y = d:
A = ∫ᶜᵈ f(y) dy
Area Between Two Curves
If f(x) ≥ g(x) on [a,b]:
A = ∫ₐᵇ [f(x) − g(x)] dx
(Upper curve minus lower curve)
A = ∫ₐᵇ [f(x) − g(x)] dx
(Upper curve minus lower curve)
Standard Areas (Frequently Asked)
Circle x² + y² = r²: Area = πr²
Ellipse x²/a² + y²/b² = 1: Area = πab
Parabola y² = 4ax: Area between vertex and latus rectum = (8a²)/3
Area between line and parabola: find intersection points, integrate the difference
Ellipse x²/a² + y²/b² = 1: Area = πab
Parabola y² = 4ax: Area between vertex and latus rectum = (8a²)/3
Area between line and parabola: find intersection points, integrate the difference
Solved Examples
Example 1
Q: Find the area enclosed by the circle x² + y² = 4.
Solution: y = √(4−x²) (upper semicircle)
Area = 4∫₀² √(4−x²) dx (by symmetry, 4 × first quadrant)
= 4 × [x√(4−x²)/2 + 2sin⁻¹(x/2)]₀² = 4 × [0 + 2(π/2)] = 4π sq units
Example 2
Q: Find the area between y = x² and y = x.
Solution: Intersection: x² = x → x = 0, 1. For 0 ≤ x ≤ 1: x ≥ x²
Area = ∫₀¹ (x − x²) dx = [x²/2 − x³/3]₀¹ = 1/2 − 1/3 = 1/6 sq units
Quick Revision Points
- Area = ∫(upper − lower)dx or ∫(right − left)dy
- Use symmetry to simplify: circle has 4-fold, ellipse has 4-fold symmetry
- Find intersection points first!
- Area is always positive — use absolute value if needed
- Common areas: circle (πr²), ellipse (πab), parabola problems
Chapter Navigation
Previous: Integrals Class 12 Notes
Next: Differential Equations Class 12 Notes
Related Chapters in Class 12 Maths
- Integrals Class 12 Notes
- Differential Equations Class 12 Notes
- Application of Derivatives Class 12 Notes
Practice What You Learned
Test yourself with our JEE Main Mock Test Set 1 to see how well you’ve mastered the concepts.