<p>\(A = 4\int_0^1 x\sqrt{1 - x^2}\, dx\). Find the value of \(A\).</p>
Step-by-Step Solution
Key Concept: Use substitution u = 1 - x² to transform the integral into a standard power function form, then evaluate the definite integral directly.
<p><strong>Step 1:</strong> Set up substitution. Let u = 1 - x², then du = -2x dx, so x dx = -½ du</p><p><strong>Step 2:</strong> Change limits: When x = 0, u = 1; when x = 1, u = 0</p><p><strong>Step 3:</strong> Substitute into integral:<br>A = 4∫₀¹ x√(1 - x²) dx = 4∫₁⁰ √u · (-½) du = 4 · ½∫₀¹ √u du = 2∫₀¹ u^(1/2) du</p><p><strong>Step 4:</strong> Evaluate: 2[u^(3/2)/(3/2)]₀¹ = 2 · (2/3)[u^(3/2)]₀¹ = (4/3)[1 - 0] = 4/3</p><p><strong>Step 5:</strong> Convert to decimal: 4/3 = 1.333... ≈ 1.33</p><p>∴ Answer: <strong>4/3 or 1.33</strong></p>
Correct Answer: 1.33