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Question #41533

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the region in the $xy$ -plane defined by $S = \left\{ (x,y) : x^2 + y^2 \le 4 \text{ and } (y - [x])(x^3 - x^2 - x + 1) \le 0 \right\}$ where $[x]$ denotes the greatest integer less than or equal to $x$ . The area of the region $S$ is:

Options

A.
$3\pi - 2$
B.
$\frac{7\pi}{3} - 2\sqrt{3}$
C.
$\frac{5\pi}{3} + \sqrt{3} - 2$
D.
$3\pi - \sqrt{3}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

The Sign-Consistency Principle|Standard Ellipse/Circle Area|Even Multiplicity Roots|Greatest Integer Function Traps

Why This Question Is Tricky

Common Mistakes

This question tests the Sign-Consistency Principle and integration over piecewise domains defined by the Greatest Integer Function (GIF).

The primary trap lies in the polynomial $(x^3 - x^2 - x + 1)$ , which factors into $(x+1)(x-1)^2$ .

Students relying on rote 'plus-minus-plus' sign alternation will incorrectly assume a sign change at the even multiplicity root $x=1$ .

This leads them to evaluate the interval $[1, 2)$ under the wrong condition $y \ge 1$ instead of $y \le 1$ , yielding the trap answer $\frac{7\pi}{3} - 2\sqrt{3}$ .

Furthermore, the GIF forces candidates to carefully partition the circular domain and compute non-trivial geometric sub-regions, filtering out those who cannot integrate circular segments bounded by off-center horizontal lines.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Integrals - Class 12
Concept	The Sign-Consistency Principle
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free