<p>The area bounded by \(y = xe^{|x|}\) and lines \(|x| = 1, y = 0\) is</p>
Step-by-Step Solution
Key Concept: The function y = xe^|x| is odd, so we can use symmetry to calculate the area on one side and double it. The region is bounded by x = -1, x = 1, and y = 0.
<p><strong>Step 1: Analyze the function</strong></p><p>Given y = xe^|x|. Check if it's odd or even:</p><p>f(-x) = (-x)e^|-x| = (-x)e^|x| = -xe^|x| = -f(x)</p><p>So f(x) is an odd function, meaning the graph is symmetric about the origin.</p><p><strong>Step 2: Determine the bounded region</strong></p><p>The region is bounded by |x| = 1 (i.e., x = -1 and x = 1), y = 0 (x-axis), and the curve y = xe^|x|.</p><p>Since the function is odd, the area below the x-axis (from x = -1 to x = 0) equals the area above the x-axis (from x = 0 to x = 1).</p><p><strong>Step 3: Calculate area for x ∈ [0, 1]</strong></p><p>For x ≥ 0, |x| = x, so y = xe^x</p><p>Area₁ = ∫₀¹ xe^x dx</p><p>Using integration by parts: Let u = x, dv = e^x dx</p><p>Then du = dx, v = e^x</p><p>∫ xe^x dx = xe^x - ∫ e^x dx = xe^x - e^x = e^x(x - 1)</p><p>Area₁ = [e^x(x - 1)]₀¹ = e¹(1 - 1) - e⁰(0 - 1) = 0 - (1)(-1) = 1</p><p><strong>Step 4: Use symmetry for total area</strong></p><p>For x ∈ [-1, 0], by odd function symmetry, the area is also 1 (but below the x-axis).</p><p>Total area = Area₁ + Area₂ = 1 + 1 = 2 sq units</p><p><strong>∴ Answer: D</strong></p>
Correct Answer: D