Area Under the Curve
Area enclosed by inverse trigonometric curves
Grade 12
Question:
<p>If the area enclosed between \(f(x) = \min\left\{\cos^{-1}(\cos x), \cot^{-1}(\cot x)\right\}\) and the x-axis in \(x \in \left(\frac{k\pi}{2}, \frac{(k+1)\pi}{2}\right)\) where \(k \in \mathbb{N}\), then k is equal to</p>
<p>(A) 4</p>
<p>(B) 6</p>
<p>(C) 8</p>
<p>(D) 12</p>
Step-by-Step Solution
Key Concept: Understand the behavior of inverse trigonometric functions and find their minimum to define the enclosed region.
<p>Analyze $\cos^{-1}(\cos x)$ and $\cot^{-1}(\cot x)$ over their respective domains. The minimum function takes the lower value in the given interval. Compute the area and match with the pattern to find k = 6.</p>
Correct Answer: B