📌 1. Introduction

Integration is not just about finding antiderivatives—it has practical applications.

👉 The main use in this chapter:

• Finding area under curves
• Area between curves

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📖 2. Area Under a Curve

Consider a function $y = f(x)$y=f(x).

The area between the curve, x-axis, and two vertical lines $x = a$x=a and $x = b$x=b is:

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$\text{Area} = \int_a^b f(x) \, dx$Area=∫ab​f(x)dx

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📌 Important Note:

• If function is above x-axis → area is positive
• If below x-axis → area is negative

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📊 3. Area Between Curve and X-axis

$$\text{Area} = \int_a^b |f(x)| dx$$Area=∫ab​∣f(x)∣dx

👉 Use modulus if curve crosses x-axis.

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📌 4. Area Between Two Curves

If curves are:

• $y = f(x)$y=f(x) (upper curve)
• $y = g(x)$y=g(x) (lower curve)

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$$\text{Area} = \int_a^b [f(x) – g(x)] dx$$Area=∫ab​[f(x)−g(x)]dx

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$\text{Area} = \int_a^b [f(x) – g(x)] dx$Area=∫ab​[f(x)−g(x)]dx

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📌 Steps:

1. Find intersection points
2. Determine upper and lower curve
3. Apply formula

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📐 5. Area with Respect to Y-axis

If curves are given as:

• $x = f(y)$x=f(y)
• $x = g(y)$x=g(y)

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$$\text{Area} = \int_c^d [f(y) – g(y)] dy$$Area=∫cd​[f(y)−g(y)]dy

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$\text{Area} = \int_c^d [f(y) – g(y)] dy$Area=∫cd​[f(y)−g(y)]dy

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📊 6. Area of Region Bounded by Curves

Common cases:

🔹 Between Line and Curve

🔹 Between Two Curves

🔹 Between Curve and Axis

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📌 7. Symmetry in Area

If function is even:$$\int_{-a}^{a} f(x) dx = 2 \int_{0}^{a} f(x) dx$$∫−aa​f(x)dx=2∫0a​f(x)dx

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$\int_{-a}^{a} f(x) dx = 2 \int_{0}^{a} f(x) dx$∫−aa​f(x)dx=2∫0a​f(x)dx

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If function is odd:$$\int_{-a}^{a} f(x) dx = 0$$∫−aa​f(x)dx=0

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📐 8. Area Using Parametric Equations

If:

• $x = f(t)$x=f(t)
• $y = g(t)$y=g(t)

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$$\text{Area} = \int y \frac{dx}{dt} dt$$Area=∫ydtdx​dt

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$\text{Area} = \int y \frac{dx}{dt} dt$Area=∫ydtdx​dt

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📊 9. Area of Standard Curves

🔹 Parabola

🔹 Circle

🔹 Ellipse

👉 Important for board exams

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📌 10. Applications in Real Life

• Finding land area
• Physics (work done)
• Economics (cost & revenue)
• Engineering problems

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❗ Common Mistakes

• Wrong limits of integration
• Not identifying upper/lower curve
• Ignoring modulus
• Sign errors

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🧠 Exam Tips

• Draw rough graphs
• Always find intersection points
• Check sign of function
• Practice NCERT questions

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📚 Practice Questions

1. Find area under curve
2. Area between two curves
3. Use symmetry
4. Solve parametric problems

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🎯 Conclusion

Application of Integrals connects calculus with geometry and real-life problems. Mastering this chapter helps in solving area-related questions efficiently and scoring high in exams.