<p>The area of the region bounded by the curves \(y = |x - 2|\), \(x = 1\), \(x = 3\) and the \(x\)-axis is</p>
Step-by-Step Solution
Key Concept: The absolute value function creates a corner at x=2, requiring you to split the integral into two regions: [1,2] where |x-2| = 2-x, and [2,3] where |x-2| = x-2.
<p><strong>Step 1:</strong> Identify where |x-2| changes sign. Since the expression (x-2) changes sign at x=2, and our interval is [1,3], we need two separate integrals.</p><p><strong>Step 2:</strong> For 1 ≤ x ≤ 2: x-2 < 0, so |x-2| = 2-x</p><p><strong>Step 3:</strong> For 2 ≤ x ≤ 3: x-2 ≥ 0, so |x-2| = x-2</p><p><strong>Step 4:</strong> Calculate the area:</p><p>Area = ∫₁² (2-x)dx + ∫₂³ (x-2)dx</p><p>= [2x - x²/2]₁² + [x²/2 - 2x]₂³</p><p>= [(4 - 2) - (2 - 1/2)] + [(9/2 - 6) - (2 - 4)]</p><p>= [2 - 3/2] + [-3/2 + 2]</p><p>= 1/2 + 1/2</p><p><strong>∴ Answer: A (Area = 1 square unit)</strong></p>
Correct Answer: A