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🌐 Chapter 8 · Class 12 Maths

Applications of Integrals

Applications of Integrals explain how integration is used to find areas under curves, volumes of solids, and other quantities. This chapter builds the basic understanding of these applications, forming a strong foundation for higher mathematics.

5
Topics
~5
Board Marks

Topics Covered

5 Topics
📖
Introduction
Meaning of Applications of Integrals · Finding Areas Under Curves
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📊
Areas Under Simple Curves
Area bounded by :-- Curves · x-axis · y-axis · lines
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🔢
Area between two Curves
Area enclosed between two curves · Functions
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➕
Integration with Graphs
Sketching graphs · Identifying regions · choosing integration limits
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📝
Properties used in area problems
Positive areas · Negative areas · Net area · symmetry
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Key Formulas

Quick Reference
Area under a curve
A = ∫ab f(x) dx
Where:
• y = f(x)
• Limits are x=a to x=b
If x = g(y)
A = ∫cd g(y) dy
Area between two curves
A = ∫ab |f(x) - g(x)| dx
If x = g(y) and x = g(y)
A = ∫cd |g(y) - h(y)| dy
Where:
• y = f(x) [ upper curve ]
• y = g(x) [ lower curve ]
Total area when curve crosses x-axis
A = ∫ab |f(x)| dx
Standard Areas
1. Circle:
    x2 + y2 = r2

  Area:

    A = πr2
2. Ellipse:
    x2/a2 + y2/b2 = 1

  Area:

    A = πab
3. Parabola:
    y2 = 4ax

  Area:

    A = 8a2/3
If area is below x-axis
Area = | ∫ab f(x)d(x)|

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