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7.Applications of Integrals (Area)
Class 12 • Mathematics • NCERT
Applications of Integrals – Area Under Curves
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This lesson explains how definite integrals are used to find the area of regions bounded by curves and lines, as prescribed in the NCERT Class 12 syllabus.
Lesson Objectives
- Understand area as an application of definite integrals.
- Find area under a curve and above x-axis.
- Calculate area between two curves.
- Solve NCERT board-level problems.
1. Area Under a Curve
The area bounded by the curve y = f(x), the x-axis, and the vertical lines x = a and x = b is given by:
Area = ∫ab f(x) dx
2. Area Above and Below x-axis
If the curve lies below the x-axis, the definite integral gives a negative value.
Area is always taken as positive value.
3. Area Between Two Curves
If y = f(x) and y = g(x) are two curves where f(x) ≥ g(x) between x = a and x = b, then:
Area = ∫ab [f(x) − g(x)] dx
4. Important Points Before Solving
Before evaluating the area, the region must be clearly identified.
• Draw rough sketch of curves
• Find points of intersection
• Decide upper and lower functions
• Find points of intersection
• Decide upper and lower functions
5. NCERT Solved Example
Find the area bounded by y = x² and x-axis between x = 0 and x = 2.
Area = ∫02 x² dx
= [x³/3]02
= 8/3 square units
6. Important NCERT Notes
• Area is always positive
• Sketching curve is compulsory
• Limits must be correct
• High-weightage board chapter
• Sketching curve is compulsory
• Limits must be correct
• High-weightage board chapter
Practice Questions (NCERT)
- Define area under a curve.
- Write formula for area between x = a and x = b.
- How is area treated when curve lies below x-axis?
- Write formula for area between two curves.
- Find area under y = x between 0 and 1.
- What is the first step before solving area problems?
- Is area ever negative?
- Find area between y = x² and x-axis from x = 1 to 2.
- Why sketching is important?
- Is this chapter important for board exams?
✅ Show Answer Key
- Area bounded by curve and axes
- ∫ab f(x) dx
- Always taken positive
- ∫ (upper − lower) dx
- 1/2
- Draw sketch
- No
- 7/3
- To identify region correctly
- Yes
© Aviate Learning – Applications of Integrals (NCERT Class 12)