Let $C_R:=\mathbb{R}\times S^1_R$ be a $2$-dimensional cylinder of radius $R$ endowed with the standard Riemannian metric $g$. Pick an arbitrary point $p\in C_R$. How to compute the volume $\operatorname{Vol}_g^R(B_1(p))$, where $B_1(p)$ denotes the ball of radius $1$ in the metric space $(C_R,d_g)$ and $d_g$ is the canonical length metric coming from $g$.
The problem I am having is to find the correct integration boundaries using the area element. I.e. Use Parametrization of cylinder and computation of area element in Lines 11-20. Then one needs to compute the integral
$$ \int_{B_1(p)} dS = \int_?^? \int_0^1 R dx dy . $$
How to proceed?