Step-by-Step Solution
Key Concept: The absolute value function creates periodic repetition. Recognize that |sin x| has period π (not 2π), and compute one period's integral, then multiply by the number of complete periods in the interval [0, 10π].
<p><strong>Step 1:</strong> Identify the period of |sin x|. Since |sin(x+π)| = |sin x|, the period is π (not 2π).</p><p><strong>Step 2:</strong> Count complete periods in [0, 10π]: There are 10π ÷ π = 10 complete periods.</p><p><strong>Step 3:</strong> Calculate one period's integral:</p><p>∫₀^π |sin x| dx = ∫₀^π sin x dx = [-cos x]₀^π = -cos π + cos 0 = -(-1) + 1 = 2</p><p><strong>Step 4:</strong> Multiply by number of periods:</p><p>∫₀^(10π) |sin x| dx = 10 × 2 = <strong>20</strong></p><p>∴ Answer: A</p>
Correct Answer: A