<p>The area of the region <em>ABCD</em> bounded by the curves <em>y</em> = <em>x</em><sup>2</sup>, <em>y</em> = 2 − <em>x</em><sup>2</sup>, and <em>x</em> = 1 (as shown in the figure) is equal to (in sq. units):</p>
Step-by-Step Solution
Key Concept: The bounded region ABCD is formed by the intersection of two parabolas y = x² and y = 2 - x², with the vertical line x = 1 acting as the right boundary. The area must be split at the intersection point where x² = 2 - x², then integrated separately over each sub-region.
<p><strong>Step 1:</strong> Find intersection of y = x² and y = 2 - x²</p><p>x² = 2 - x² ⟹ 2x² = 2 ⟹ x = 1 (taking positive value)</p><p>At x = 1: y = 1</p><p><strong>Step 2:</strong> Identify the bounded region ABCD</p><p>For 0 ≤ x ≤ 1, the region is bounded above by y = 2 - x² and below by y = x²</p><p><strong>Step 3:</strong> Set up the integral</p><p>Area = ∫₀¹ [(2 - x²) - x²] dx = ∫₀¹ (2 - 2x²) dx</p><p><strong>Step 4:</strong> Evaluate the integral</p><p>= [2x - (2x³)/3]₀¹</p><p>= 2(1) - (2·1³)/3 - 0</p><p>= 2 - 2/3</p><p>= 6/3 - 2/3 = 4/3 ≈ 1.33</p><p><strong>Note:</strong> If answer is 1.01, the region ABCD may have additional geometric constraints (such as involving x = 0 or another boundary). With standard interpretation: <strong>∴ Answer: 4/3 sq. units (or verify problem statement for 1.01)</strong></p>
Correct Answer: 1.01