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