= Solution
{c}
Compute the <line integral> on each straight segment. On the first, $y=z=0$ and only $dx$ contributes; on the second, $x=1,z=0$ and only $dy$ contributes; on the third, $x=y=1$ and only $dz$ contributes. Therefore
$$
\begin{aligned}
\int_C\mathbf F\cdot d\mathbf r
&=\int_0^1 3x^2\,dx+\int_0^1 0\,dy+\int_0^1 20z^2\,dz\\
&=1+0+\frac{20}{3}=\boxed{\frac{23}{3}}.
\end{aligned}
$$
Although $F_x=9$ on the second segment, that term does no <work> there because $dx=0$.
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