Given the partition $\{0,\,\frac{\pi}{4},\,\frac{\pi}{2},\,\frac{3\pi}{4},\,\pi\}$, and points $\bar{x}_1=\frac{\pi}{8}$, $\bar{x}_2=\frac{3\pi}{8}$, $\bar{x}_3=\frac{5\pi}{8}$, and $\bar{x}_4=\frac{7\pi}{8}$; find $R(P)$ for $\int_0^{\pi}\sin(x)\,dx$.

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