Reference for an integral's convergence on an $n$-ball when $n>2$. I was searching for a reference of a standard result from calculus. Unfortunately I couldn't find it. I  think that's mostly due to I am not familiar with any english calculus book. 
So I am searching for the following:
For $$h=(h_1,\ldots,h_n)\in\mathbb{R}^n,$$ 
the integral
$$\int_{\|\boldsymbol h\|<\epsilon} \frac{d\boldsymbol h}{\|\boldsymbol h\|^2}$$
is finite for $n>2$ and does not exist for $n=1,2$, where the norm is arbitrary. 
Does anyone knows a reference for this result?
Can anyone recommend an English calculus book (by calculus I mean differentiation, integration, measure theory, etc)?
Thank you very much,
Analyst
 A: For fixed $\epsilon$ and arbitrary dimension $n$, $\|\mathbf{h}\|^2$ is radially symmetric with respect to the origin, if we assume your norm is an $l^p$-norm:
$$\|\mathbf{h}\| = \left(\sum_{i=1}^n h_i^p\right)^{1/p}, \quad \text{where }p>0.$$
Denote your integration domain (a ball in $p$-norm ) by 
$$B(0;\epsilon) = \{\mathbf{h}:\|\mathbf{h}\|< \epsilon \}.$$
Using the integration transform to the spherical coordinates: 
$$
\int_{ B(0;\epsilon)}f\,dx = \int_0^{\epsilon}\left\{\int_{\partial B(0;r)} f\,dS\right\}\,dr,\tag{1}
$$
where $f = 1/\|\mathbf{h}\|^2 = 1/r^2$ on the $(n-1)$-sphere $\partial B(0;r)$, hence (1) is 
$$
\int_0^{\epsilon}\frac{1}{r^2}\left\{\int_{\partial B(0;r)} 1\,dS\right\}\,dr = \int_0^{\epsilon}\frac{1}{r^2} \mathrm{meas}\{\partial B(0;r)\}\,dr.
$$
The surface area of $\partial B(0;r)$ is $\omega(n,p) r^{n-1}$ where $\omega(n,p)$ is a constant, which is the surface area of the $l^p$-unit sphere, and it is related to the dimension $n$ and $p$. Now above becomes
$$
\omega(n,p)\int_0^{\epsilon}\frac{1}{r^2} r^{n-1} \,dr=\omega(n,p)\int_0^{\epsilon}r^{n-3}\,dr.
$$
This integral diverges for $n=1,2$. For $n\geq 3$ it clearly converges.
If you are asking about a reference, (1) can be derived from coarea formula. For the surface area of the $l^p$-unit sphere, the formula can be derived from the volume of $l^p$-unit balls following the first link's recursive relation.
