<p>The area enclosed between the curves \(y^2 = x\) and \(y = |x|\) is</p>
Step-by-Step Solution
Key Concept: The curves y² = x (parabola opening right) and y = |x| (V-shaped) intersect at (0,0) and (1,1), (1,-1). Due to symmetry about the x-axis, calculate the area in the upper half and double it, integrating with respect to y to avoid absolute value complications.
<p><strong>Step 1:</strong> Find intersection points. From y² = x and y = |x|:</p><p>For y ≥ 0: y² = x and y = x gives y² = y, so y(y-1) = 0, yielding y = 0 or y = 1.</p><p>Intersection points: (0, 0) and (1, 1). By symmetry, also (1, -1).</p><p><strong>Step 2:</strong> Express x in terms of y. From the curves: x = y² (parabola) and x = y (line for y ≥ 0).</p><p>For 0 ≤ y ≤ 1: the line x = y is to the right of parabola x = y².</p><p><strong>Step 3:</strong> Calculate area in upper half:</p><p>A₁ = ∫₀¹ (y - y²) dy = [y²/2 - y³/3]₀¹ = 1/2 - 1/3 = 1/6</p><p><strong>Step 4:</strong> By symmetry about x-axis, total area = 2 × 1/6 = <strong>1/3</strong></p><p>∴ Answer: D</p>
Correct Answer: D