Class 12 Maths Applications of Integrals Notes

8.1 Introduction

Integrals are not just abstract concepts—they are widely used to measure quantities like area, volume, and displacement. This chapter focuses on practical applications of definite integrals.

• Helps in calculating area under curves
• Used in physics, economics, and engineering problems
• Builds on knowledge from Chapter 7: Integrals

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8.2 Area under Simple Curves

The area under a curve y = f(x) between x = a and x = b is given by the definite integral:$$\text{Area} = \int_a^b f(x) \, dx$$Area=∫ab​f(x)dx

🔹 Cases:

1. Curve above x-axis

$$f(x) \ge 0 \quad \Rightarrow \quad \text{Area} = \int_a^b f(x) dx$$f(x)≥0⇒Area=∫ab​f(x)dx

2. Curve below x-axis

$$f(x) \le 0 \quad \Rightarrow \quad \text{Area} = -\int_a^b f(x) dx$$f(x)≤0⇒Area=−∫ab​f(x)dx

3. Curve crossing x-axis

• Split the interval at points where f(x) = 0
• Calculate area above and below separately
• Take absolute values before adding

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🔹 Area between Two Curves

If y = f(x) and y = g(x) are two curves with f(x) ≥ g(x) for x ∈ [a, b]:$$\text{Area} = \int_a^b [f(x) – g(x)] \, dx$$Area=∫ab​[f(x)−g(x)]dx

• Represents the total area enclosed between the curves.

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🔹 Important Notes

• Definite integrals give net area, so negative values indicate the curve is below x-axis
• Splitting intervals ensures correct total area calculation
• Often used in geometry, physics, and economics problems

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✅ Key Points to Remember

• Area under a curve = definite integral
• If curve crosses axis, calculate area in segments
• Area between two curves = difference of integrals
• Applications appear in real-life problems like distance, work, and economics