<p>Evaluate <span>\(\int_1^2 [x^3 - 1] \, dx\)</span>, where <span>\([\cdot]\)</span> denotes the greatest integer function.</p>
Step-by-Step Solution
Key Concept: Split the integral at points where the greatest integer function changes value, then evaluate each piece as a simple rectangular area.
<p><strong>Solution:</strong></p><p>For <span>$1 \le x \le 2$</span>, we have <span>$1 \le x^3 \le 8$</span>, so <span>$0 \le x^3 - 1 \le 7$</span>.</p><p>The integral is split into intervals based on where <span>$[x^3 - 1]$</span> takes constant values:</p><ul><li>When <span>$x \in [1, 2^{1/3})$</span>, we have <span>$x^3 - 1 \in [0, 1)$</span>, so <span>$[x^3 - 1] = 0$</span></li><li>When <span>$x \in [2^{1/3}, 3^{1/3})$</span>, we have <span>$x^3 - 1 \in [1, 2)$</span>, so <span>$[x^3 - 1] = 1$</span></li><li>When <span>$x \in [3^{1/3}, \ldots, 2]$</span>, the pattern continues</li></ul><p>Therefore, <span>$I = \int_1^{2^{1/3}} 0 \, dx + \int_{2^{1/3}}^{3^{1/3}} 1 \, dx + \ldots$</span></p><p>∴ The answer is (d) None of these.</p>
Correct Answer: D