<p>The area (in sq. units) of the region enclosed by the curves <br> \(C_1: y + 2x^2 = 0\) and \(C_2: y + 3x^2 = 1\) is</p>
Step-by-Step Solution
Key Concept: Find intersection points of the two curves, then integrate the difference of the upper and lower curves. The area between curves is ∫|f(x) - g(x)|dx over the intersection interval.
<p><strong>Step 1: Find intersection points</strong></p><p>From C₁: y = -2x²</p><p>From C₂: y = 1 - 3x²</p><p>Setting equal: -2x² = 1 - 3x²</p><p>x² = 1 → x = ±1</p><p><strong>Step 2: Determine which curve is above</strong></p><p>At x = 0: C₁ gives y = 0, C₂ gives y = 1</p><p>So C₂ is above C₁ in the interval [-1, 1]</p><p><strong>Step 3: Set up the integral</strong></p><p>Area = ∫₋₁¹ [(1 - 3x²) - (-2x²)] dx</p><p>= ∫₋₁¹ (1 - 3x² + 2x²) dx</p><p>= ∫₋₁¹ (1 - x²) dx</p><p><strong>Step 4: Evaluate</strong></p><p>= [x - x³/3]₋₁¹</p><p>= (1 - 1/3) - (-1 + 1/3)</p><p>= 2/3 - (-2/3)</p><p>= 4/3 sq. units</p><p>∴ Answer: <strong>4/3</strong></p>
Correct Answer: C