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AIEEE 2002

JEE Main 2002 Mathematics question on Definite Integrals and Areas

The area (in sq units) bounded by the curves y=∣x∣−1y = |x| - 1 and y=−∣x∣+1y = -|x| + 1 is
  1. A.
    1
  2. B.
    2
  3. C.
    222\sqrt{2}
  4. D.
    4
Show answer and solution

Correct option: B

Solution

The two curves form a rhombus with vertices at (0,1), (1,0), (0,-1), (-1,0). By symmetry, compute the area for x≥0: upper curve y = -x+1, lower curve y = x-1, they intersect at x=0 and x=1. Area for x≥0 = ∫₀¹ [(-x+1)-(x-1)]dx = ∫₀¹ (2-2x)dx = [2x-x²]₀¹ = 1. By symmetry the left half also gives 1, so total area = 2.

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