<p>Evaluate the following: <br> 30. \(\int_0^{2\pi} \sqrt{2ax - x^2} \, dx\)</p>
Step-by-Step Solution
Key Concept: Complete the square inside the radical to convert √(2ax - x²) into the form √(r² - (x - h)²), which represents a semicircle. This allows geometric interpretation or trigonometric substitution.
<p><strong>Step 1:</strong> Complete the square inside the radical:</p><p>2ax - x² = -(x² - 2ax) = -(x² - 2ax + a²) + a² = a² - (x - a)²</p><p><strong>Step 2:</strong> Rewrite the integral:</p><p>∫₀²ᵖ √(a² - (x - a)²) dx</p><p><strong>Step 3:</strong> Recognize geometric interpretation. Let u = x - a, then du = dx. When x = 0, u = -a; when x = 2π, u = 2π - a.</p><p>However, note the original limits go from 0 to 2π. For a > 0, the expression a² - (x - a)² ≥ 0 requires |x - a| ≤ a, i.e., 0 ≤ x ≤ 2a.</p><p><strong>Step 4:</strong> The integrand √(a² - (x - a)²) represents a semicircle of radius a centered at x = a. The integral from x = 0 to x = 2a gives the area of a complete semicircle:</p><p>Area = (1/2)πa²</p><p><strong>Step 5:</strong> For the full limits 0 to 2π (assuming the integral is only non-zero from 0 to 2a):</p><p>If the question means 0 to 2a: ∫₀²ᵃ √(a² - (x - a)²) dx = <strong>πa²/2</strong></p><p>If interpreted as 0 to 2π with specific a value, the answer depends on the relationship between a and π.</p><p><strong>Most likely answer:</strong> <strong>πa²/2</strong> (assuming limits should be 0 to 2a, or a = π giving πa²/2 = π³/2)</p>
Correct Answer: 0