📘 Applications of Integration – Class 12 (NCERT Detailed Summary)
🔹 1. Introduction
The chapter Applications of Integration deals with the practical use of integrals to find areas of regions bounded by curves and straight lines 📐. It helps convert abstract integration concepts into visual and real-life interpretations.
🔹 2. Area Under a Curve
If y = f(x) is a continuous function and is non-negative in the interval [a, b], then the area bounded by the curve, the x-axis, and the vertical lines x = a and x = b is given by:
Area = ∫ab f(x) dx 🧮
This concept is based on dividing the region into very small strips and summing their areas using integration.
🔹 3. Area Above the x-axis and Below the x-axis
When a curve lies below the x-axis, the definite integral gives a negative value. However, area is always taken as positive ➕.
Area between the curve and x-axis:
- Above x-axis: ∫ f(x) dx
- Below x-axis: −∫ f(x) dx
🔹 4. Area Between Two Curves
To find the area enclosed between two curves y = f(x) and y = g(x) between x = a and x = b, where f(x) ≥ g(x):
Area = ∫ab [f(x) − g(x)] dx ✨
This method is useful in solving problems involving intersecting curves.
🔹 5. Area Between a Curve and Lines Parallel to Axes
Sometimes integration is done with respect to y instead of x. If x = f(y), then:
Area = ∫ [xright − xleft] dy
This approach is helpful when curves are easier to express in terms of y.
🔹 6. Steps to Find Area Using Integration
- Draw a rough sketch of the region ✏️
- Identify limits of integration 🎯
- Decide whether to integrate with respect to x or y
- Apply the correct formula and simplify
🔹 7. Important Observations
- Area is always a positive quantity 📏
- Correct limits are crucial for accurate results ⚠️
- Graphical understanding simplifies problem-solving 📈
🌟 Conclusion
The chapter Applications of Integration builds a strong foundation for solving geometrical and physical problems using calculus 🌍. Mastering this chapter is essential for scoring well in board exams and higher studies 💯.