Area between two curves

This section goes beyond the rationalised NCERT chapter; JEE Main still sets these questions.

If f(x) ≥ g(x) on [a, b], a vertical strip between the curves has height f(x) − g(x), so the area between them is A = ∫ₐᵇ [f(x) − g(x)] dx. It does not matter whether the curves are above or below the axis.

The limits are usually the x-coordinates where the curves meet: solve f(x) = g(x).

Worked example: y = x and y = x² meet at x = 0 and x = 1, and x ≥ x² between them. The area is ∫₀¹ (x − x²) dx = 1/2 − 1/3 = 1/6.

Worked example: y = x + 2 and y = x² meet at x = −1 and x = 2. The area is ∫₋₁² (x + 2 − x²) dx = 9/2.

Worked example: y² = 4x and x² = 4y meet at (0, 0) and (4, 4). Between them 2√x ≥ x²/4, so the area is ∫₀⁴ (2√x − x²/4) dx = 32/3 − 16/3 = 16/3.

With horizontal strips the rule becomes A = ∫꜀ᵈ [x_right − x_left] dy. For example, between x = y² and x = y + 2, ∫₋₁² (y + 2 − y²) dy = 9/2.

If the curves cross inside the interval, the one on top changes. Split at each crossing and use (upper − lower) on each piece.