<p>Evaluate \(\int_{-20\pi}^{20\pi} |\cos x|\,dx\).</p>
Step-by-Step Solution
Key Concept: Use the periodicity of |cos x| with period π to reduce the integral into manageable chunks. |cos x| has period π (not 2π), so the integral over 40π can be expressed as 40 times the integral over one period.
<p><strong>Step 1:</strong> Recognize that |cos x| has period π (since cos(x + π) = -cos x, so |cos(x + π)| = |cos x|).</p><p><strong>Step 2:</strong> The interval [-20π, 20π] has length 40π, which equals exactly 40 periods of |cos x|.</p><p><strong>Step 3:</strong> Calculate ∫₀^π |cos x| dx. Split into [0, π/2] and [π/2, π]:</p><p>∫₀^(π/2) cos x dx = [sin x]₀^(π/2) = 1</p><p>∫_(π/2)^π |cos x| dx = ∫_(π/2)^π (-cos x) dx = [-sin x]_(π/2)^π = 0 - (-1) = 1</p><p>So ∫₀^π |cos x| dx = 1 + 1 = 2</p><p><strong>Step 4:</strong> By periodicity: ∫₋₂₀π^(20π) |cos x| dx = 40 × ∫₀^π |cos x| dx = 40 × 2 = 80</p><p>∴ Answer: <strong>80</strong></p>
Correct Answer: 80