I wonder if the following holds true: Let $z:[0,1]\times B^{n-1}_r(0)\to(M^n,g), (t,p)\mapsto z(t,p)$ be a diffeomorphism a.e. onto its image with respect to the Lebesgue measure on $A:=[0,1]\times B^{n-1}_r(0)$.
I have two questions:
(1) Is there a variant of the co-area formula that gives me the following identity which I expect to be true
$$ \int_{z(A)}\mathrm d\operatorname{vol}_g = \int_{B^{n-1}(0)}\mathcal H^1_g(z([0,1]\times\{p\}))\mathrm d\mathcal L^{n-1}(p), $$ where $\mathcal H^1_g$ denotes the 1-Hausdorff measure with respect to $g$ and $\mathcal L^{n-1}$ is the $(n-1)$-dimenaional Lebesgue measure? If yes, does someone have a reference or a proof?
(2) How can one show that $$\mathcal H^1_g(z([0,1]\times\{p\}))=\int_0^1\sqrt{g\left(\frac{\partial z}{\partial s}(s,p),\frac{\partial z}{\partial s}(s,p)\right)}~\mathrm ds,$$ which I expect also to be true.