Definite Integration
Comparison of definite integrals
Grade 12
Question:
<p>If \(I_1 = \int_{0}^{1} 2x^2 \, dx\), \(I_2 = \int_{0}^{1} 2x^3 \, dx\), \(I_3 = \int_{1}^{2} 2x^2 \, dx\) and \(I_4 = \int_{1}^{2} 2x^3 \, dx\), then</p>
<p>\(I_2 > I_1\)</p>
<p>\(I_1 > I_2\)</p>
<p>\(I_3 = I_4\)</p>
<p>\(I_3 > I_4\)</p>
Step-by-Step Solution
Key Concept: Compare definite integrals by evaluating them explicitly using the fundamental theorem of calculus, then order them by comparing the resulting numerical values.
<p><strong>Step 1: Evaluate I₁</strong></p><p>I₁ = ∫₀¹ 2x² dx = [2x³/3]₀¹ = 2/3</p><p><strong>Step 2: Evaluate I₂</strong></p><p>I₂ = ∫₀¹ 2x³ dx = [2x⁴/4]₀¹ = 1/2</p><p><strong>Step 3: Evaluate I₃</strong></p><p>I₃ = ∫₁² 2x² dx = [2x³/3]₁² = (16/3 - 2/3) = 14/3</p><p><strong>Step 4: Evaluate I₄</strong></p><p>I₄ = ∫₁² 2x³ dx = [2x⁴/4]₁² = (8 - 1/2) = 15/2</p><p><strong>Step 5: Compare values</strong></p><p>I₁ = 2/3 ≈ 0.667</p><p>I₂ = 1/2 = 0.5</p><p>I₃ = 14/3 ≈ 4.667</p><p>I₄ = 15/2 = 7.5</p><p>∴ Order: I₂ < I₁ < I₃ < I₄</p>
Correct Answer: B