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OpenArea bounded by curves and lines using definite integrals, including areas between two curves.
Area bounded by curves and lines using definite integrals, including areas between two curves.
Maths · Class 12 · Integral Calculus
Area Under Curves for JEE
Everything you need for this topic on one page: the ideas, the formulas, the traps, and free lectures that explain it well.
Last updated: 26 October 2026
Key ideas
Area Under Curves: what you must understand
- Sketch the curves first
- Area is always positive: integrate |f|
- Between curves: ∫(upper − lower) dx
- Sometimes integrating in y is easier
Formula sheet
Must-know formulas for Area Under Curves
- A = ∫ₐᵇ |f(x)| dx
- Between curves: A = ∫ₐᵇ (f − g) dx where f ≥ g
- Area of ellipse = πab
- Area between y² = 4ax and x² = 4ay is 16a²/3
Print
One-page formula sheet
Printable Area Under Curves formula sheet: formulas, key ideas and traps on one page. Save it as a PDF for offline revision.
Avoid
Common mistakes students make
- Not finding intersection points
- Negative area from below the x-axis
- Choosing the wrong upper curve
For the exam
How this topic is asked in JEE
Appears frequently. Almost every shift has an area question. A quick sketch is the whole trick.
Prerequisites: Definite Integration.
NCERT map
Where Area Under Curves is in NCERT
Chapter names follow the current rationalised NCERT textbooks. Read these sections before solving PYQs.
| Class | Book | Chapter | Sections JEE asks from |
|---|---|---|---|
| Class 12 | Mathematics Part II | Application of Integrals | Area under Simple Curves; Area of a Region Bounded by a Curve and a Line; Area between Two Curves |
JEE Advanced level
A JEE Advanced-level example on Area Under Curves
Question. Find the area of the region defined by {(x, y) : x² + y² ≤ 4, x + y ≥ 2}. This is the region inside the circle of radius 2 centered at the origin that lies above the line x + y = 2.
Show solution
- Find intersection points of circle x² + y² = 4 and line x + y = 2. Substitute y = 2 - x into circle equation: x² + (2-x)² = 4, which gives 2x² - 4x = 0, so x = 0 or x = 2. Points are (0, 2) and (2, 0).
- The chord from (0, 2) to (2, 0) cuts off a circular segment. The required area is this segment (the smaller piece above the line).
- Find the angle subtended at center. Point (2, 0) is at angle 0 and point (0, 2) is at angle π/2. So the sector angle θ = π/2.
- Area of circular sector = (1/2)r²θ = (1/2)(4)(π/2) = π.
- Area of triangle with vertices at origin, (0, 2), and (2, 0) = (1/2) × base × height = (1/2)(2)(2) = 2.
- Required area = Area of sector - Area of triangle = π - 2.
Answer: π - 2 square units
This example was solved twice independently and both solutions agree.
Free lectures
Best free lectures for Area Under Curves
Full-chapter free lectures picked from YouTube search results by views, from established JEE channels (mostly in Hinglish). Use the subtopic list to jump straight to the part you need (timestamps come from each video's own chapter list). Watch one, then solve PYQs the same day.
- Area Under Curve | One Shot | #BounceBack Series | Unacademy Atoms | JEE Maths | Nishant Vora — JEE Legends, 3:37:12
- AREA UNDER CURVE in One Shot: All Concepts & PYQs Covered | JEE Main & Advanced — JEE Wallah, 3:07:32
Practise
Where to practise for free
- 8 JEE Main 2026 PYQs on Area Under Curves, with solutions.
- Read the chapter in NCERT textbooks (official, free), then solve its NCERT Exemplar (official, free) questions.
- Chapter-wise past questions: JEE Main and Advanced PYQs and question bank.
- Timed practice tests: NTA Abhyas practice tests (official, free) and our free mock tests.
- See it move: interactive simulation.
FAQs
Frequently asked questions
Is Area Under Curves important for JEE Main 2027?
Appears frequently. It is part of the official 'Integral Calculus' unit in the JEE Main 2027 syllabus.
What are the most important formulas in Area Under Curves?
A = ∫ₐᵇ |f(x)| dx; Between curves: A = ∫ₐᵇ (f − g) dx where f ≥ g; Area of ellipse = πab.
How should I study Area Under Curves for JEE?
Understand the key ideas from NCERT or one full lecture, learn the formula sheet, then solve 25–30 past JEE questions on the topic. Almost every shift has an area question. A quick sketch is the whole trick.