Consider the warped product $(0, \infty) \times \mathbb{S}^{n-1}$ with the metric given by $$g = dr^2 + (\phi(r))^2 g_{\mathbb{S}^{n-1}}$$
I want to compute the volume of a ball of radius $s$ centered at the origin in this space. The book I'm reading claims it is given by:
$$ \operatorname{Vol}(B(O, s))=n \omega_{n} \int_{0}^{s} \phi(r)^{n-1} d r \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1) $$ where $n\omega_{n}$ denotes the volume form of the unit euclidean $(n-1)$-sphere. Further, it states that under the hypotheses
$C^{-1} r^{1 / 2} \leq \phi(r) \leq C r^{1 / 2}$, $\phi^{\prime}(r)=O\left(r^{-1/2}\right)$ and $\phi^{\prime \prime}(r)=O\left(r^{-3 / 2}\right)$ and for $s \geq 1$, we will have that:
$$ \operatorname{Vol}(B(O, s)) \approx \int_{0}^{s} r^{\frac{n-1}{2}} d r \approx s^{\frac{n+1}{2}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2) $$
So it would appear that:
$$ \operatorname{Vol}(B(O, s))= \int_{\mathbb{S}^{n-1}} \int_{0}^{s} dV$$
where $dV$ would be the volume form of this warped product. But if this is correct, why is it so (and if not, how can I get to the formula presented)? And how do I compute this volume form (why does it apparently equal $\phi(r)^{n-1}$?)? I'm lost on how to compute it for a warped product. Any help here would be very appreciated (and please, give as many details as possible).
My other question is: how can one go from $(1)$ to $(2)$? I don't see how the hypotheses mentioned allow me to approximate $\phi$ in this manner.
UPDATE: Following Ted Shifrin's suggestion, I tried the cases $n = 2$ and $n = 3$. For $n = 2$, since the metric on $\mathbb{S}^1$ is indistinguishable from that of $\mathbb{R}$, it's easy to compute that $g$ is given by the matrix
$$ \left(\begin{array}{ccc}{1} & {0} \\ {0} & {(\phi(r))^2}\end{array}\right) $$
for $n = 3$, we get:
$$ \left(\begin{array}{ccc}{1} & {0} & {0} \\ {0} & {(\phi(r))^2} & {0} \\ {0} & {0} & {(\phi(r))^2 \sin^2(\theta)} \\ \end{array}\right) $$
It was then easy to see that for the general case, the matrix for $g$ will be given by:
$$ \left(\begin{array}{ccc}{1} & {} \\ {} & {(\phi(r))^2 g_{\mathbb{S}^{n-1}}}\end{array}\right) $$
where the unfilled spaces correspond to zeroes. So the volume of a ball would be given by:
$$ \operatorname{Vol}(B(O, s))= \int_{\mathbb{S}^{n-1}} \int_{0}^{s} dV$$
but $dV = \sqrt{\operatorname{det} g }\ dx_1 \ \cdots dx_{n-1} \ dr$ and since $r$ is independent of all the other $x_1, \cdots, x_{n-1}$ and $\sqrt{\operatorname{det} g } = \sqrt{\operatorname{det}(\phi(r))^2 g_{\mathbb{S}^{n-1}}} = (\phi(r))^{n-1}$, we have exactly:
$$ \operatorname{Vol}(B(O, s))=n \omega_{n} \int_{0}^{s} \phi(r)^{n-1} d r \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $$
as for the other question, I think it deserves another different post to solve it.