GATE EC 2022 Set 1 Q55 — Vector Analysis

In an electrostatic field, the electric displacement density vector, $\overrightarrow{D},$ is given by

$$\overrightarrow{D} (x, y, z) = (x^{3} \overrightarrow{i} + y^{3} \overrightarrow{j} + xy^{2} \overrightarrow{k}) \; \text{C/m}^{2},$$

where $\overrightarrow{i}, \overrightarrow{j}, \overrightarrow{k}$ are the unit vectors along $\text{x-axis, y-axis,}$ and $\text{z-axis},$ respectively. Consider a cubical region $R$ centered at the origin with each side of length $1 \; \text{m},$ and vertices at $(\pm 0.5 \; \text{m,} \pm 0.5 \; \text{m,} \pm 0.5 \; \text{m}).$ The electric charge enclosed within $R$ is _____________ $\text{C}$ (rounded off to two decimal places).