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Question #40880

Difficulty: 8/10 | 🟢 Free

The Question

Let $V$ be the region in $\mathbb{R}^3$ defined by $|x| + |y| + |z| \le 2$ . Evaluate the integral: $I = \iiint_V \left( x^2 \{x\} + y^2 \{y\} + z^2 \{z\} \right) \,dx\,dy\,dz$ where $\{t\}$ denotes the fractional part of $t$ .

Options

A.
$\frac{32}{5}$
B.
$\frac{64}{5}$
C.
0
D.
$-\frac{32}{5}$

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

Octant Sign Convention|Axis of Integration Selection|Integral Part and Fractional Part

Why This Question Is Tricky

Common Mistakes

This question tests the evaluation of volume integrals involving the fractional part function over a symmetric 3D region.

The primary trap lies in the parity of the fractional part function $\{x\}$ .

Students frequently and incorrectly assume that $\{x\}$ is an odd function, which would lead them to erroneously conclude that the integral evaluates to zero over the symmetric domain.

Furthermore, when converting $\{x\}$ to $x - [x]$ , students often mishandle the greatest integer function for negative values, forgetting that $[-x] = -1 - [x]$ for non-integers.

The problem elegantly rewards the application of the identity $\int_{-a}^a f(x)[x]dx = -\int_0^a f(x)dx$ for even functions, bypassing tedious piecewise integration across different octants while rigorously testing the sign convention of step functions.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Introduction to Three-dimensional Geometry - Class 11
Concept	Octant Sign Convention
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free