Answer
$\int_C \textbf F \cdot d\textbf r = 0$
Work Step by Step
$\textbf r(t) = 3\sin t \textbf i + 3 \cos t \textbf j = x(t) \textbf i + y(t) \textbf j$
It follows that $x(t) = 3\sin t , y(t) = 3 \cos t$
$\textbf r' = 3\cos t \textbf i - 3\sin t \textbf j \\
\textbf F(x,y) = x\textbf i + y \textbf j\\
\Rightarrow \textbf F (x(t),y(t))= 3\sin t \textbf i + 3 \cos t \textbf j\\
\int_C \textbf F \cdot d\textbf r\\
= \int_C \textbf F \cdot \textbf r' dt\\
= \int ( 3\sin t \textbf i + 3 \cos t \textbf j)\cdot (3\cos t \textbf i - 3\sin t \textbf j ) dt\\
= \int (9 \sin t \cos t - 9 \sin t \cos t) dt\\
=0$
(proved)
