= Solution
On the first segment, $x=z=0$, so both <line integrals> vanish. On the second, $y=1,z=0$, giving contributions $\int_0^23x^2dx=8$ and $\int_0^22x\,dx=4$. On the last, $x=2,y=1$, the contributions are $\int_0^3-2z\,dz=-9$ and $\int_0^3-6z^2dz=-54$. Therefore
$$
\boxed{\int\boldsymbol F_1\cdot d\boldsymbol r=-1,\qquad\int\boldsymbol F_2\cdot d\boldsymbol r=-50.}
$$
The first also equals $\phi(B)-\phi(A)=8-9=-1$, as the <fundamental theorem for line integrals> predicts.
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