I am trying to calculate $\iint\limits_S \overset{\rightharpoonup} F \cdot \overset{\rightharpoonup} n\ dS$ using the divergence theorem.
It is given that $\overset{\rightharpoonup} F = x \hat{i} + y \hat{j} +z \hat{k}$ and $V$ is a 3-dimensional object having volume $4$ bounded by the surface $S$.
Here are the steps I have taken:
Divergence theorem
$$\iint\limits_S \overset{\rightharpoonup} F \cdot \overset{\rightharpoonup} n\ dS = \iiint\limits_V \text{div} \overset{\rightharpoonup} F dx\ dy\ dz$$
Find the divergence of $\overset{\rightharpoonup} F$
$$\text{div} \overset{\rightharpoonup} F = \frac{\partial}{\partial x}x + \frac{\partial}{\partial y}y + \frac{\partial}{\partial z}z = 3$$
Substitute into the divergence theorem
$$\iiint\limits_V (3)\ dx\ dy\ dz$$
$$3 \cdot \iiint\limits_V\ dx\ dy\ dz$$
Given the volume is 4,
$$\iint\limits_S \overset{\rightharpoonup} F \cdot \overset{\rightharpoonup} n\ dS = 3 \cdot 4 = 12$$
Is this correct?
My main doubt is whether it is permissible to replace $\iiint\limits_V\ dx\ dy\ dz$ with $4$ in the last step.