Step-by-Step Solution
Key Concept: Recognize that 1 - x² changes sign at x = 1 within [0,1]. Since 1 - x² ≥ 0 for all x ∈ [0,1], the absolute value can be removed directly, simplifying the integral to ∫₀¹ (1 - x²) dx.
<p><strong>Step 1:</strong> Determine the sign of (1 - x²) on [0,1].</p><p>For x ∈ [0,1]: x² ∈ [0,1], so 1 - x² ∈ [0,1] ≥ 0.</p><p>Therefore, |1 - x²| = 1 - x² on the entire interval.</p><p><strong>Step 2:</strong> Evaluate the integral.</p><p>∫₀¹ |1 - x²| dx = ∫₀¹ (1 - x²) dx</p><p><strong>Step 3:</strong> Find the antiderivative.</p><p>= [x - x³/3]₀¹</p><p><strong>Step 4:</strong> Apply limits.</p><p>= (1 - 1/3) - (0 - 0) = 2/3</p><p>∴ Answer: B (2/3)</p>
Correct Answer: B