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Question #41482

Difficulty: 8/10 | 🟢 Free

The Question

Let $R = \left\{ (x, y) \in \mathbb{R}^2 : -10 \le y \le 14, \min(f(y), g(y)) \le x \le \max(f(y), g(y)) \right\}$ , where $f(y) = \text{sgn}(y) \sqrt{\{y\} - \{y\}^2}$ and $g(y) = \frac{1}{2} - \left| \{y\} - \frac{1}{2} \right|$ . Here, $\{y\}$ denotes the fractional part of $y$ and $\text{sgn}(y)$ denotes the signum function. The area of the region $R$ is:

Options

A.
$3\pi - 1$
B.
$3\pi - 6$
C.
$3\pi + 6$
D.
$3\pi - 2$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Horizontal vs Vertical Strips|Symmetry and Periodicity|Signum Function Discontinuity

Why This Question Is Tricky

Common Mistakes

This question tests the strategic choice of integration axis and the handling of signum-induced asymmetries.

Attempting to integrate with respect to $x$ (vertical strips) forces the inversion of periodic fractional part functions, creating an unsolvable analytic nightmare.

The intended path uses horizontal strips (integrating w.r.t $y$ ).

The primary trap lies in the $\text{sgn}(y)$ function combined with asymmetric limits ( $-10$ to $14$ ).

For $y > 0$ , the semi-circle and triangle are on the same side of the y-axis, so their areas subtract.

For $y < 0$ , the signum function flips the semi-circle to the negative x-axis, causing the areas to add.

Students who blindly assume symmetry and multiply the $y > 0$ area by 24 will incorrectly obtain $3\pi - 6$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Integrals - Class 12
Concept	Horizontal vs Vertical Strips
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free