Chapter 8 — Application of Integrals

Chapter 8 of the Class 12 Maths NCERT textbook, "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.

• Areas as accumulated strips — The chapter's central idea is to build an area by summing infinitely thin strips: vertical strips integrated as ∫y dx, or horizontal strips as ∫x dy. This lets students measure regions that have no simple geometric formula.
• Handling curves above and below the axis — When a curve dips below the x-axis its integral turns negative, so the chapter uses absolute values to keep areas positive — splitting a region into parts where needed, as with the area under y = cos x over a full period.
• Recovering classic area formulas — By integrating standard curves the chapter re-derives familiar results — the circle's πa² and the ellipse's πab — using symmetry to integrate over one quadrant and scale up, showing calculus underpins the geometry formulas students already know.

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.