<p>Evaluate: \[I = \int_{-1.5}^{3.5} |x-1|\, dx\]</p>
Step-by-Step Solution
Key Concept: Split the integral at the point where the absolute value expression changes sign (x=1), then evaluate each piece separately using the definition of absolute value.
<p><strong>Step 1:</strong> Identify where |x-1| changes sign. The expression inside equals zero at x=1.</p><p><strong>Step 2:</strong> Split the integral at x=1:</p><p>I = ∫₍₋₁.₅₎¹ |x-1| dx + ∫₁³·⁵ |x-1| dx</p><p><strong>Step 3:</strong> For x ∈ [-1.5, 1): x-1 < 0, so |x-1| = -(x-1) = 1-x</p><p>∫₍₋₁.₅₎¹ (1-x) dx = [x - x²/2]₍₋₁.₅₎¹ = (1 - 1/2) - (-1.5 - 2.25/2)</p><p>= 0.5 - (-1.5 - 1.125) = 0.5 - (-2.625) = 3.125</p><p><strong>Step 4:</strong> For x ∈ [1, 3.5]: x-1 ≥ 0, so |x-1| = x-1</p><p>∫₁³·⁵ (x-1) dx = [x²/2 - x]₁³·⁵ = (12.25/2 - 3.5) - (1/2 - 1)</p><p>= (6.125 - 3.5) - (-0.5) = 2.625 + 0.5 = 3.125</p><p><strong>Step 5:</strong> I = 3.125 + 3.125 = 6.25</p><p>∴ Answer: <strong>6.25</strong></p>
Correct Answer: 6.25