<p>The area of the region bounded by the curves \(y = |x - 1|\) and \(y = 3 - |x|\) is</p>
Step-by-Step Solution
Key Concept: Identify intersection points by solving |x - 1| = 3 - |x| in different regions (x < 0, 0 ≤ x < 1, x ≥ 1), then integrate the difference of upper and lower curves over the bounded region.
<p><strong>Step 1: Analyze the absolute value functions piecewise</strong></p><p>For y = |x - 1|: y = 1 - x (x < 1) and y = x - 1 (x ≥ 1)</p><p>For y = 3 - |x|: y = 3 + x (x < 0) and y = 3 - x (x ≥ 0)</p><p><strong>Step 2: Find intersection points</strong></p><p>For x < 0: 1 - x = 3 + x → 2x = -2 → x = -1, y = 2</p><p>For 0 ≤ x < 1: 1 - x = 3 - x → 1 = 3 (no solution)</p><p>For x ≥ 1: x - 1 = 3 - x → 2x = 4 → x = 2, y = 1</p><p>Intersection points: (-1, 2) and (2, 1)</p><p><strong>Step 3: Determine which function is above</strong></p><p>At x = 0: |0 - 1| = 1 and 3 - |0| = 3, so y = 3 - |x| is above</p><p><strong>Step 4: Calculate the area</strong></p><p>Area = ∫₋₁⁰ [(3 + x) - (1 - x)] dx + ∫₀² [(3 - x) - (x - 1)] dx</p><p>= ∫₋₁⁰ (2 + 2x) dx + ∫₀² (4 - 2x) dx</p><p>= [2x + x²]₋₁⁰ + [4x - x²]₀²</p><p>= (0 - (-2 + 1)) + ((8 - 4) - 0)</p><p>= 1 + 4 = 5</p><p>∴ Answer: C</p>
Correct Answer: C