<p>The value of \(\int_{0}^{2} (x^2) dx \left\{ \int_{a}^{b} f(x) dx = \int_{a}^{c} f(x)dx + \int_{c}^{b} f(x)dx \right\}\) where <span>\(f(x) = [x^2]\)</span> (greatest integer function) is equal to:</p>
<p>\(-\sqrt{2} - \sqrt{3} + 5\)</p>
<p>\(-\sqrt{2} + \sqrt{3} + 5\)</p>
<p>\(\sqrt{2} + \sqrt{3} - 5\)</p>
<p>\(\sqrt{2} - \sqrt{3} + 5\)</p>
Step-by-Step Solution
Key Concept: Break the integral at points where [x²] changes value. Since [x²] is a step function, identify where x² crosses integer boundaries (at x=1, x=√2, x=√3, x=√4=2) within [0,2], then integrate the constant value on each subinterval.
<p><strong>Step 1:</strong> Identify where [x²] changes value on [0,2]:</p><ul><li>When 0 ≤ x < 1: x² ∈ [0,1), so [x²] = 0</li><li>When 1 ≤ x < √2: x² ∈ [1,2), so [x²] = 1</li><li>When √2 ≤ x < √3: x² ∈ [2,3), so [x²] = 2</li><li>When √3 ≤ x ≤ 2: x² ∈ [3,4], so [x²] = 3</li></ul><p><strong>Step 2:</strong> Apply additivity of definite integrals:</p><p>∫₀² [x²]dx = ∫₀¹ 0·dx + ∫₁^√2 1·dx + ∫_{√2}^{√3} 2·dx + ∫_{√3}² 3·dx</p><p><strong>Step 3:</strong> Calculate each integral:</p><ul><li>∫₀¹ 0·dx = 0</li><li>∫₁^√2 1·dx = (√2 - 1)</li><li>∫_{√2}^{√3} 2·dx = 2(√3 - √2)</li><li>∫_{√3}² 3·dx = 3(2 - √3)</li></ul><p><strong>Step 4:</strong> Sum all parts:</p><p>= 0 + (√2 - 1) + 2√3 - 2√2 + 6 - 3√3</p><p>= √2 - 1 + 2√3 - 2√2 + 6 - 3√3</p><p>= 5 - √2 - √3</p><p>∴ Answer: A</p>
Correct Answer: A