<p>Evaluate \(\displaystyle\int_{-1}^{1.5} [x^2]\, dx\), where \([x]\) is the greatest integral (floor) function.</p>
Step-by-Step Solution
Key Concept: Split the integral at points where ⌊x²⌋ changes value (i.e., where x² crosses integer values), then evaluate ⌊x²⌋ as a constant on each subinterval.
<p><strong>Step 1:</strong> Identify where ⌊x²⌋ changes. We need x² = 0, 1, 2, ... within [-1, 1.5].</p><p>x² = 0 ⟹ x = 0</p><p>x² = 1 ⟹ x = ±1</p><p>x² = 2 ⟹ x = ±√2 ≈ ±1.414</p><p>Critical points in [-1, 1.5]: x = -1, 0, 1, √2</p><p><strong>Step 2:</strong> Determine ⌊x²⌋ on each subinterval:</p><p>• On [-1, 0): x² ∈ [0, 1), so ⌊x²⌋ = 0</p><p>• On [0, 1): x² ∈ [0, 1), so ⌊x²⌋ = 0</p><p>• On [1, √2): x² ∈ [1, 2), so ⌊x²⌋ = 1</p><p>• On [√2, 1.5]: x² ∈ [2, 2.25], so ⌊x²⌋ = 2</p><p><strong>Step 3:</strong> Compute the integral:</p><p>∫₋₁^1.5 ⌊x²⌋ dx = ∫₋₁^0 0 dx + ∫₀^1 0 dx + ∫₁^√2 1 dx + ∫_{√2}^1.5 2 dx</p><p>= 0 + 0 + 1(√2 - 1) + 2(1.5 - √2)</p><p>= √2 - 1 + 3 - 2√2</p><p>= 2 - √2</p><p>∴ Answer: <strong>2 - √2</strong> (or approximately <strong>0.586</strong>)</p>
Correct Answer: 2