Area Under the Curve
Area Bounded by Multiple Curves
Grade 12
Question:
<p>The area of region formed by points <i>(x, y)</i> satisfying <i>x</i> + <i>y</i> ≤ 6, <i>x</i><sup>2</sup> + <i>y</i><sup>2</sup> ≤ 6<i>y</i> and <i>y</i><sup>2</sup> ≤ 8<i>x</i> is equal to <i>kπ</i> − 2/12, then <i>k</i> =</p>
Step-by-Step Solution
Key Concept: Identify the three boundary curves, find their intersection points, and compute the area of the region satisfying all three inequalities simultaneously.
<p><strong>Solution:</strong> The three conditions define:</p><p>(1) <i>x</i> + <i>y</i> ≤ 6: Region below line <i>x</i> + <i>y</i> = 6</p><p>(2) <i>x</i><sup>2</sup> + <i>y</i><sup>2</sup> ≤ 6<i>y</i>: Circle <i>x</i><sup>2</sup> + (<i>y</i> − 3)<sup>2</sup> ≤ 9 (center (0,3), radius 3)</p><p>(3) <i>y</i><sup>2</sup> ≤ 8<i>x</i>: Parabola opening rightward</p><p>Finding the intersection region and computing the area yields: Area = (27π − 2)/12</p><p>Therefore, <i>k</i> = 27.</p><p>∴ Answer is <strong>Q (27)</strong>.</p>
Correct Answer: Q