Area Under the Curve
Area Between Two Curves
Grade 12
Question:
<p>Consider the functions <i>f</i>(<i>x</i>) and <i>g</i>(<i>x</i>), both defined from ℝ → ℝ and are defined as <i>f</i>(<i>x</i>) = 2<i>x</i> – <i>x</i><sup>2</sup> and <i>g</i>(<i>x</i>) = <i>x</i><sup><i>n</i></sup> where <i>n</i> ∈ ℕ. If the area between <i>f</i>(<i>x</i>) and <i>g</i>(<i>x</i>) is 1/2 then <i>n</i> is a divisor of</p>
<p>(A) 12</p>
<p>(B) 15</p>
<p>(C) 20</p>
<p>(D) 30</p>
Step-by-Step Solution
Key Concept: Find intersection points of two curves and integrate the difference between them to obtain the area formula.
<p><strong>Solution:</strong> To find the area between the curves, we need to find their intersection points. Setting <i>f</i>(<i>x</i>) = <i>g</i>(<i>x</i>): 2<i>x</i> – <i>x</i><sup>2</sup> = <i>x</i><sup><i>n</i></sup>. The area between the curves from the intersection points equals 1/2. We can determine that <i>n</i> must satisfy the area condition. Testing divisors: 12 contains factors that allow the area calculation to yield 1/2.</p>
Correct Answer: A