Answer
$\int_C \textbf F \cdot d\textbf r = 0$
Work Step by Step
$\textbf r(t) = t \textbf i - t^3 \textbf j = x(t) \textbf i + y(t) \textbf j$
It follows that $x(t) = t , y(t) = -t^3$
$\therefore \textbf r' = \textbf i -3t^2 \textbf j \\
\textbf F(x,y) = -3y\textbf i + x \textbf j\\
\Rightarrow \textbf F (x(t),y(t))= 3t^3 \textbf i + t \textbf j\\
\int_C \textbf F \cdot d\textbf r\\
= \int_C \textbf F \cdot \textbf r' dt\\
= \int ( 3t^3 \textbf i + t \textbf j )\cdot ( \textbf i -3t^2 \textbf j ) dt\\
= \int (3t^3-3t^3) dt\\
=0$
(proved)
