Area Under the Curve
Area Between Curved and Linear Boundaries
Grade 12
Question:
<p>The area enclosed between the curves \(|x| + |y| \geq 2\) and \(y^2 = 4\left(1 - \frac{x^2}{9}\right)\) is:</p>
<p>(a) \((6\pi - 4)\) sq. units</p>
<p>(b) \((6\pi - 8)\) sq. units</p>
<p>(c) \((3\pi - 4)\) sq. units</p>
<p>(d) \((3\pi - 2)\) sq. units</p>
Step-by-Step Solution
Key Concept: Recognize the constraint as an ellipse with semi-major axis 3 and semi-minor axis 2, and subtract the area of overlap with the diamond region $|x| + |y| = 2$.
<p>The region $|x| + |y| \geq 2$ is the exterior of a square diamond with vertices at (±2, 0) and (0, ±2). The curve $y^2 = 4\left(1 - \frac{x^2}{9}\right)$ is an ellipse. The area enclosed between them equals the area of the ellipse minus the area of intersection with the diamond region.</p><p>∴ Answer is (b).</p>
Correct Answer: B