Solution
= Solution
{c}
Parametrize the straight segment by $\mathbf r(t)=(t,t,t)$, $0\leq t\leq1$. Then the <line integral> is
$$
\begin{aligned}
\int_C\mathbf F\cdot d\mathbf r
&=\int_0^1\bigl(3t^2+6t-14t^2+20t^3\bigr)\,dt\\
&=3-\frac{11}{3}+5=\boxed{\frac{13}{3}}.
\end{aligned}
$$
The three different answers are possible because this <vector field> is not a <conservative vector field>; for example $\partial_yF_x=6$ whereas $\partial_xF_y=0$.