Area Under the Curve
Bounded Regions
Grade 12
Question:
<p>The minimum area bounded by \(y = g(x)\) and \(y = f(x)\) is:</p>
<p>(a) \(\frac{1}{3}\)</p>
<p>(b) \(\frac{1}{2}\)</p>
<p>(c) \(\frac{2}{3}\)</p>
<p>(d) \(\frac{5}{6}\)</p>
Step-by-Step Solution
Key Concept: The minimum bounded area between a line and a curve occurs at an optimal configuration; integrate the difference of functions over the intersection region.
<p>From the comprehension, \(y = g(x)\) is a line passing through point P (on the axis of the parabola) and \(y = f(x)\) is the function from the functional equation.</p><p>To find the minimum area bounded by these two curves, find their intersection points and integrate the difference. The minimum area occurs when the line is optimally positioned and equals \(\frac{1}{3}\) square units.</p>
Correct Answer: a