= Solution
The <parametric curve> runs from $t=0$ to $t=1$, with $\boldsymbol r'=(2,3t^2,6t)$. Substitution into the two <line integrals> gives
$$
\boldsymbol F_1(\boldsymbol r(t))\cdot\boldsymbol r'=48t^5-90t^9,\qquad
\boldsymbol F_2(\boldsymbol r(t))\cdot\boldsymbol r'=20t^4-351t^6.
$$
Hence
$$
\boxed{\int\boldsymbol F_1\cdot d\boldsymbol r=-1,\qquad\int\boldsymbol F_2\cdot d\boldsymbol r=-\frac{323}7.}
$$
The first is unchanged because the field is <conservative>. The two different answers for the second exhibit its path dependence, consistent with its nonzero <curl>.
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