I am looking for a reference - a published book - of this fact
the volume of the ball of small radius $r > 0$ in normal coordinates around point $p\in M$, a smooth Riemannian manifold, equals
$$Vol(B_{r}(p)) = ω_{m}r^{m} (1 − \frac{S(p)}{6(m + 2)}r^{2} + O(|r|^3)),$$
where $\omega_{m}$ is the volume of the unit ball in $\mathrm{R}^{m}$, $m$ is the dimension of $M$, and $S(p)$ is the constant curvature at point $p$.