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NDA 2025 · PYQ · Area under curve · medium

Let the function f(x) = x² − 1. What is the area bounded by the function f(x) and the x-axis?

  1. A.1/3 square unit
  2. B.2/3 square unit
  3. C.4/3 square units✓ Correct
  4. D.2 square units

Explanation

f(x) = x² − 1 crosses the x-axis at x = ±1. Between these, f(x) ≤ 0. Area = ∫₋₁¹ |x² − 1| dx = ∫₋₁¹ (1 − x²) dx = [x − x³/3]₋₁¹ = (1 − 1/3) − (−1 + 1/3) = 2/3 + 2/3 = 4/3 square units.
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