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Question #41532

Difficulty: 8/10 | 🟢 Free

The Question

Evaluate the definite integral: $\int_{0}^{2} \left| |x^2 - 2| - 1 \right| \, dx$

Options

A.
$4\sqrt{3} - \frac{8\sqrt{2}}{3} - 2$
B.
$4\sqrt{3} - \frac{10}{3}$
C.
$2\sqrt{3} - 2$
D.
$4\sqrt{3} - \frac{4\sqrt{2}}{3} - 2$

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

The Sign-Consistency Principle|Fundamental Theorem of Calculus|Modulus Function Breakdown|The Modulus Trap

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to rigorously apply the Sign-Consistency Principle when integrating nested modulus functions.

The primary trap lies in failing to identify all critical points—specifically the cusp at $x = \sqrt{2}$ generated by the inner modulus $|x^2 - 2|$ .

Students often only solve for the roots of the outer function ( $x=1, \sqrt{3}$ ) and erroneously integrate across $\sqrt{2}$ without flipping the sign of the inner argument.

This leads to calculating the 'net' area instead of the 'total' area, directly penalizing the confusion between integrating an absolute value and taking the absolute value of an integral.

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