<p>The value of $\int_{3/4}^{2} |1 - x^2| dx$ is</p>
Step-by-Step Solution
Key Concept: Since |1 - x²| changes sign at x = 1, we must split the integral at this critical point and evaluate the absolute value expression separately on each interval.
<p><strong>Step 1: Find where |1 - x²| changes sign</strong></p><p>Set 1 - x² = 0 ⟹ x = ±1</p><p>In our interval [3/4, 2]:</p><ul><li>At x = 3/4: 1 - (3/4)² = 1 - 9/16 = 7/16 > 0, so |1 - x²| = 1 - x²</li><li>At x = 2: 1 - 4 = -3 < 0, so |1 - x²| = x² - 1</li><li>Critical point x = 1 lies in [3/4, 2]</li></ul></p><p><strong>Step 2: Split the integral at x = 1</strong></p><p>∫₃/₄² |1 - x²| dx = ∫₃/₄¹ (1 - x²) dx + ∫₁² (x² - 1) dx</p></p><p><strong>Step 3: Evaluate the first integral</strong></p><p>∫₃/₄¹ (1 - x²) dx = [x - x³/3]₃/₄¹</p><p>= (1 - 1/3) - (3/4 - (3/4)³/3)</p><p>= 2/3 - (3/4 - 27/64·1/3)</p><p>= 2/3 - (3/4 - 9/64)</p><p>= 2/3 - (48/64 - 9/64)</p><p>= 2/3 - 39/64</p><p>= 128/192 - 117/192 = 11/192</p></p><p><strong>Step 4: Evaluate the second integral</strong></p><p>∫₁² (x² - 1) dx = [x³/3 - x]₁²</p><p>= (8/3 - 2) - (1/3 - 1)</p><p>= (8/3 - 6/3) - (1/3 - 3/3)</p><p>= 2/3 - (-2/3)</p><p>= 2/3 + 2/3 = 4/3</p></p><p><strong>Step 5: Add the results</strong></p><p>Total = 11/192 + 4/3 = 11/192 + 256/192 = 267/192 = 89/64</p><p><strong>∴ Answer: 89/64</strong></p>
Correct Answer: 89