= Solution
Interpret the axis-directed route as successive coordinate-axis-parallel segments
$$
(0,0,0)\longrightarrow(1,0,0)\longrightarrow(1,2,0)\longrightarrow(1,2,1).
$$
On the first segment, $y=z=0$, so the field vanishes. On the second, $z=0$, so it again vanishes. On the last, $x=1$, $y=2$ and only $dz$ is nonzero, giving $\mathbf F\cdot d\mathbf r=8z\,dz$. Thus
$$
\boxed{\int_C\mathbf F\cdot d\mathbf r=0+0+\int_0^1 8z\,dz=4}.
$$
The second and third segments are parallel to the corresponding global axes; they do not return to the global $y$- or $z$-axis before reaching the endpoint.
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