= Solution
Parametrize the segment in its stated direction by $\mathbf x(t)=(1-t,t,t)$, $0\le t\le1$. Then $d\mathbf x=(-1,1,1)\,dt$, and substitution into the two <vector fields> gives
$$
\mathbf F(\mathbf x(t))=(2t,1,1),\qquad
\mathbf G(\mathbf x(t))=(2t^2,1-2t+2t^2,1-2t+2t^2).
$$
Taking each <dot product> with $d\mathbf x$ evaluates the <line integrals>:
$$
\boxed{\int\mathbf F\cdot d\mathbf x=\int_0^1(2-2t)\,dt=1,\qquad
\int\mathbf G\cdot d\mathbf x=\int_0^1(2-4t+2t^2)\,dt=\frac23.}
$$
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