<p>Evaluate \[I = \int_0^{10\pi} |\sin x|\,dx\]</p>
Step-by-Step Solution
Key Concept: The absolute value function creates a periodic pattern. Since |sin x| has period π (not 2π), we can count complete periods in [0, 10π] and use the integral of sin x over one period [0, π].
<p><strong>Step 1:</strong> Recognize that |sin x| has period π (since sin x alternates between positive and negative every π units).</p><p><strong>Step 2:</strong> Count complete periods in [0, 10π]: Number of periods = 10π/π = 10 complete periods.</p><p><strong>Step 3:</strong> Calculate ∫₀^π |sin x| dx. Since sin x ≥ 0 on [0, π]:
∫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> Use periodicity:
I = ∫₀^{10π} |sin x| dx = 10 × ∫₀^π |sin x| dx = 10 × 2 = 20</p><p>∴ Answer: <strong>20</strong></p>
Correct Answer: A