For each positive integer $n$, define a function $f _ { n }$ on $[ 0, 1 ]$ as follows:
$$f _ { n } ( x ) = \left\{ \begin{array} { c c c }
0 & \text { if } & x = 0 \\
\sin \frac { \pi } { 2 n } & \text { if } & 0 < x \leq \frac { 1 } { n } \\
\sin \frac { 2 \pi } { 2 n } & \text { if } & \frac { 1 } { n } < x \leq \frac { 2 } { n } \\
\sin \frac { 3 \pi } { 2 n } & \text { if } & \frac { 2 } { n } < x \leq \frac { 3 } { n } \\
\vdots & \vdots & \vdots \\
\sin \frac { n \pi } { 2 n } & \text { if } & \frac { n - 1 } { n } < x \leq 1
\end{array} \right.$$
Then, the value of $\lim _ { n \rightarrow \infty } \int _ { 0 } ^ { 1 } f _ { n } ( x ) d x$ is\\
(A) $\pi$\\
(B) 1\\
(C) $\frac { 1 } { \pi }$\\
(D) $\frac { 2 } { \pi }$