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Question #41573

Difficulty: 8/10 | 🟢 Free

The Question

Let $R = \{(x,y) \in \mathbb{R}^2 : x \ge 0, 0 \le y \le x^2\}$ . Evaluate the integral $I = \iint_R \text{sgn}(x-y) e^{-x} \,dx\,dy$ where $\text{sgn}(\cdot)$ denotes the signum function.

Options

A.
$2 - \frac{6}{e}$
B.
$2 - \frac{4}{e}$
C.
$2$
D.
$\text{The integral diverges}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Region Ambiguity|Unbounded Region Trap|Signum Function Discontinuity

Why This Question Is Tricky

Common Mistakes

This question tests the 'Region Ambiguity' and 'Unbounded Region Trap'.

Students often assume an unbounded region with an exponential and polynomial mix might diverge, or they mishandle the jump discontinuity of the signum function across $y=x$ .

The trap lies in failing to correctly partition the unbounded region $y \le x^2$ at the intersection with the discontinuity line $y=x$ , leading to incorrect bounds or a false conclusion of divergence.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Integrals - Class 12
Concept	Region Ambiguity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free