The value of the definite integral ∈t_-3^3 ∈t_-2^2 ∈t_-1^1 ≤ft(4x^2y - z^3) dz\, dy\, dx is _______. (Rounded off to the nearest integer)

GATE CSE 2023 Q31: Triple integral of 4x²y − z³ over a symmetric box

The value of the definite integral

$\int_{-3}^{3} \int_{-2}^{2} \int_{-1}^{1} \left(4x^2y - z^3\right) dz\, dy\, dx$

is _______. (Rounded off to the nearest integer)

Try it: numerical answer: type the value

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Numerical answer. No negative marking.

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Answer: 0 (official GATE 2023 final answer key)

Concept used: An odd function integrated over an interval symmetric about 0 gives 0.

First step: Note that $4x^2y$ is odd in $y$ and $z^3$ is odd in $z$ , and the limits are symmetric.

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Source: official GATE 2023 CS question paper and final answer key, organised by IIT Kanpur. Question paper · Answer key