Answer
FALSE
Work Step by Step
Using Green's theorem, we know that
\begin{align*}
\oint_C y\, dx &= \iint\limits_D \left( \frac{ \partial (0) }{ \partial x } - \frac{ \partial (y) }{ \partial y } \right)\, dA\\
&= \iint\limits_D -1\, dA\\
&\neq \iint\limits_D 1\, dA\\
&= A
\end{align*}
where $D$ is the region bounded by $C$.
Thus, the statement is false.