Area between two curves: Class 12 practice
The area between y = f(x) (upper) and y = g(x) (lower) from a to b is ∫ₐᵇ [f(x) − g(x)] dx. Read the method, follow the three worked examples, then try the six questions.
What you should know first
- Definite integrals
The idea behind area between two curves
The area between y = f(x) (upper) and y = g(x) (lower) from a to b is ∫ₐᵇ [f(x) − g(x)] dx. Find where the curves meet to get the limits.
Step-by-step method
- Solve f(x) = g(x) for the limits.
- Decide which curve is on top.
- Integrate (top − bottom).
Worked example 1 (Basic)
Find the area enclosed by y = x² and y = 2x.
- Meet at x = 0 and x = 2
- ∫₀^2 (2x − x²) dx = 2³/6
- Answer: 4/3
Worked example 2 (Standard)
Find the area under y = x² from x = 0 to x = 6.
- ∫₀^6 x² dx = 6³/3
- Answer: 72
Worked example 3 (Challenge)
Find the area enclosed between y = 25 − x² and the x-axis.
- Roots x = ±5
- ∫ (25 − x²) dx from −5 to 5 = 4·5³/3
- Answer: 500/3
Common mistakes and misconceptions
- Subtracting in the wrong order
- Guessing the limits
Practice questions
- (Basic) Find the area enclosed by y = x² and y = 3x.
- (Basic) Find the area enclosed by y = x² and y = 6x.
- (Standard) Find the area under y = x² from x = 0 to x = 2.
- (Standard) Find the area under y = x² from x = 0 to x = 4.
- (Challenge) Find the area enclosed between y = 16 − x² and the x-axis.
- (Challenge) Find the area enclosed between y = 4 − x² and the x-axis.
Answer key with full working
- Q1: Meet at x = 0 and x = 3 → ∫₀^3 (3x − x²) dx = 3³/6 → Answer: 9/2
- Q2: Meet at x = 0 and x = 6 → ∫₀^6 (6x − x²) dx = 6³/6 → Answer: 36
- Q3: ∫₀^2 x² dx = 2³/3 → Answer: 8/3
- Q4: ∫₀^4 x² dx = 4³/3 → Answer: 64/3
- Q5: Roots x = ±4 → ∫ (16 − x²) dx from −4 to 4 = 4·4³/3 → Answer: 256/3
- Q6: Roots x = ±2 → ∫ (4 − x²) dx from −2 to 2 = 4·2³/3 → Answer: 32/3
Syllabus note
This skill is listed under "Application of Integrals" for Class 12 in the NCERT/CBSE books. ICSE, IGCSE, IB and state boards may teach it in a different chapter or year.
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.
Related practice and notes
Frequently asked questions
Which chapter is area between two curves in?
"Application of Integrals", Class 12 Maths (NCERT/CBSE). Other boards may place it differently.
How were the answers checked?
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.