Solution
= Solution
For the parabola, $x=t$, $y=1+t^2$, $dx=dt$, and $dy=2t\,dt$. Hence
$$
\begin{aligned}
\int_0^1\left[-1+
2t\bigl((1+t^2)^2+t\bigr)\right]dt
&=\int_0^1(-1+2t+2t^2+4t^3+2t^5)\,dt\\
&=\boxed{2}.
\end{aligned}
$$
The difference from the straight-path value reflects that the one-form is not exact.