<p>If \(m = \displaystyle\int_{-2}^{0} \frac{|\sin x|}{\left[\dfrac{x}{\pi}\right] + \dfrac{1}{2}}\,dx\) and \(n = \displaystyle\int_0^2 \frac{|\sin x|}{\left[\dfrac{x}{\pi}\right] + \dfrac{1}{2}}\,dx\), where [ ] is the greatest integer function, then \(-\dfrac{m}{n}\) = ______.</p>
Step-by-Step Solution
Key Concept: The greatest integer function [x/π] is constant on specific intervals, allowing you to evaluate it separately for negative and positive domains. Use symmetry properties of |sin x| and substitution to relate the two integrals.
<p><strong>Step 1: Analyze [x/π] on each domain</strong></p><p>For x ∈ [-2, 0]: x/π ∈ [-2/π, 0] ≈ [-0.637, 0], so [x/π] = -1</p><p>For x ∈ [0, 2]: x/π ∈ [0, 2/π] ≈ [0, 0.637], so [x/π] = 0</p><p><strong>Step 2: Simplify denominators</strong></p><p>For m: [x/π] + 1/2 = -1 + 1/2 = -1/2</p><p>For n: [x/π] + 1/2 = 0 + 1/2 = 1/2</p><p><strong>Step 3: Express integrals</strong></p><p>m = ∫₋₂⁰ |sin x|/(-1/2) dx = -2∫₋₂⁰ |sin x| dx</p><p>n = ∫₀² |sin x|/(1/2) dx = 2∫₀² |sin x| dx</p><p><strong>Step 4: Use substitution in m</strong></p><p>Let u = -x in the integral for m. When x ∈ [-2, 0], u ∈ [0, 2]</p><p>Since sin(-x) = -sin(x), we have |sin(-x)| = |sin(x)|</p><p>So ∫₋₂⁰ |sin x| dx = ∫₀² |sin u| du</p><p><strong>Step 5: Calculate ratio</strong></p><p>m = -2∫₀² |sin x| dx = -2 · (same integral in n's coefficient)</p><p>n = 2∫₀² |sin x| dx</p><p>Therefore: -m/n = -(-2∫₀² |sin x| dx)/(2∫₀² |sin x| dx) = 2/2 = 1</p><p>∴ Answer: <strong>1</strong></p>
Correct Answer: 1