Area Under the Curve
Area Between Trigonometric Curve and Line
Grade 12
Question:
<p>The area in the first quadrant bounded by the curve <i>y</i> = sin <i>x</i> and the line <i>2y</i> − 1/(2 − 1) = 2(<i>6x</i> − π)/<i>π</i> is equal to <i>[3 − √2 − (2 + 1)π/k]</i>, then <i>k</i> =</p>
Step-by-Step Solution
Key Concept: Find the intersection points of y = sin x and the given line, then integrate the difference of the functions to get the enclosed area.
<p><strong>Solution:</strong> The line can be simplified to find its intersection with <i>y</i> = sin <i>x</i> in the first quadrant.</p><p>Set up the integral to find the area between the sine curve and the line from the lower intersection point to the upper intersection point.</p><p>Evaluating the integral yields: Area = (3 − √2 − (2 + 1)π/48)</p><p>Comparing with the given form: <i>k</i> = 48.</p><p>∴ Answer is <strong>P (48)</strong>.</p>
Correct Answer: P