Chapter 8 — Application of Integrals

NCERT Class 12 Maths Chapter 8, Application of Integrals, teaches how to use definite integrals to find areas bounded by curves such as circles, ellipses, parabolas, and lines — going beyond the area formulas of elementary geometry.

Chapter 8 of NCERT Class 12 Mathematics Part II covers the Application of Integrals, focusing on finding areas under and between curves using definite integrals. Students learn to compute areas bounded by curves like y = f(x) and the x-axis using vertical strips (∫ y dx) or horizontal strips (∫ x dy). The chapter covers standard curves including circles, ellipses, and trigonometric functions, with worked examples showing that the area of a circle x² + y² = a² equals πa² and the area of an ellipse x²/a² + y²/b² = 1 equals πab. When a curve dips below the x-axis, the absolute value of the integral is used.

1. 01Area under a curve y = f(x) between x = a and x = b is given by A = ∫ₐᵇ f(x) dx using vertical strips of width dx.
2. 02Area bounded by x = g(y), the y-axis, and lines y = c and y = d is given by A = ∫꜀ᵈ g(y) dy using horizontal strips.
3. 03If a curve lies below the x-axis, the area is taken as the absolute value of the definite integral, since f(x) < 0 gives a negative result.
4. 04The area enclosed by the circle x² + y² = a² is πa², derived by integrating over one quadrant and multiplying by 4.
5. 05The area enclosed by the ellipse x²/a² + y²/b² = 1 is πab, obtained similarly using the symmetry of the ellipse.
6. 06The area bounded by y = cos x between x = 0 and x = 2π equals 4, computed by summing absolute areas of the three regions above and below the x-axis.