I have two questions (somewhat related) regarding local geometry on a SMOOTH, COMPACT Riemannian manifold. I still have a hard time getting a "good" understanding of local geometry.

Question 1:

It is true that there exists $\epsilon>0$ such that for all $r < \epsilon$, there exists $c_g >0$ (indep. of $x\in M$) such that

$$ c_g r^n \leq Vol_g (B_x(r)), \forall x \in M, $$

where $B_x(r)$ is the geodesic ball centered at $x$. In other words, is it true that small geodesic ball are "comparable" to Euclidean balls.

Question 2:

I am trying to compute an integral on a "small" geodesic ball, namely the following:

$$ I = \int_{(exp_{x_0}(B_{0}(R))} \rho(x_0,x)^{2-n} dV_g$$

Using normal coordinates at $x_0$, we should have that

\begin{eqnarray*} I & =& \int_{(exp_{x_0}(B_{0}(R))} \rho(x_0,x)^{2-n} dV_g. \\ & \leq & C_g \int_0^{R} \rho^{2-n} \rho^{n-1} (1 + O(\rho^2)) d\rho \\ \end{eqnarray*}

I really need to get that last inequality but I am very unsure about it. I think it should be true, at least for small enough $R$ (hopefully for $R$ smaller than the $\epsilon$ defined in question 1.

Is that possible? Any feedback would be appreciated.