<p>Find the area bounded by the curve <br>\(y = 2x - x^2\) and the straight line \(y = -x\).</p>
Step-by-Step Solution
Key Concept: Find intersection points of the two curves by solving 2x - x² = -x, then integrate the difference (upper curve minus lower curve) between these limits.
<p><strong>Step 1: Find intersection points</strong></p><p>Set y = 2x - x² equal to y = -x:</p><p>2x - x² = -x</p><p>3x - x² = 0</p><p>x(3 - x) = 0</p><p>x = 0 or x = 3</p><p><strong>Step 2: Determine which curve is above</strong></p><p>At x = 1 (test point between 0 and 3):</p><p>Parabola: y = 2(1) - 1² = 1</p><p>Line: y = -1</p><p>The parabola y = 2x - x² is above the line y = -x</p><p><strong>Step 3: Set up and evaluate the integral</strong></p><p>Area = ∫₀³ [(2x - x²) - (-x)] dx</p><p>= ∫₀³ (3x - x²) dx</p><p>= [3x²/2 - x³/3]₀³</p><p>= 3(3)²/2 - (3)³/3 - 0</p><p>= 27/2 - 9</p><p>= 27/2 - 18/2</p><p>∴ Answer: <strong>9/2</strong></p>
Correct Answer: 9/2