NCERT Solutions for Class 12 Mathematics Chapter 8 Application of Integrals
NCERT Solutions for Class 12 Mathematics Chapter 8 Application of Integrals covers using definite integrals to find the area under a curve, the area enclosed between two curves, and the areas of standard geometric shapes such as circles and ellipses.
Overview of Topics & Subtopics (Click to Jump)
1. Area Under a Simple Curve
Short Note:
• The area enclosed between a curve y = f(x), the x-axis, and the ordinates x = a and x = b is given by the definite integral ∫[a to b] f(x)dx (taking the absolute value if the curve dips below the x-axis).
• When the required region lies along the y-axis, the area is instead computed by integrating x as a function of y between the corresponding limits.
2. Area Between Two Curves
Short Note:
• The area enclosed between two curves y = f(x) and y = g(x) over an interval is found by integrating the difference between the upper and lower functions, [f(x) - g(x)], over that interval.
• The points of intersection of the two curves, found by solving their equations simultaneously, determine the limits of integration for the enclosed region.
3. Areas of Standard Curves
Short Note:
• The area of a circle of radius r, given by x² + y² = r², can be derived using integration and matches the well-known geometric formula πr².
• Similarly, the area enclosed by an ellipse (x²/a² + y²/b² = 1) can be found using integration, giving the formula πab.
NCERT Textbook Exercise Questions
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