Help Calculating a certain integral I study an article, and I got stuck on a problem of calculating an integral. Whatever I do, I do not get the result mentioned there. The notations are $u,\tilde u$ are functions defined on $\Omega \subset \Bbb{R}^N$ with values in $\Bbb{R}^n$:
$$ \eta_\varepsilon = \int_\Omega \tilde u(x)dx -\int_\Omega u(x)dx \in \Bbb{R}^n $$
The idea is that we try to correct $\tilde u$ such that it has the same integral as $u$. So pick a ball $B_\varepsilon=B(x_0,\varepsilon^{1/N})$ contained in the interior of $\Omega$. We define the function 
$$ v(x)=\begin{cases} \tilde u(x) & x \notin B_\varepsilon \\
 \tilde u(x)+h_\varepsilon(1-\varepsilon^{-1/N}|x-x_0|) & x \in B_\varepsilon \end{cases}$$
where $h_\varepsilon \in \Bbb{R}^n$ should be chosen such that $\int_\Omega v=\int_\Omega u$, i.e. 
$$ \int_{B_\varepsilon} h_\varepsilon(1-\varepsilon^{-1/N}|x-x_0|)dx=-\eta_\varepsilon $$
Since $h_\varepsilon$ is a constant, it should be enough to find the value of 
$$(I) \ \ \ \ \int_{B_\varepsilon} (1-\varepsilon^{-1/N}|x-x_0|)dx $$
which I calculated with the coarea formula and got 
$$ |B_\varepsilon|-\varepsilon^{-1/N}\int_0^{\varepsilon^{1/N}}r^N \cdot N \omega_N dr=\frac{ \varepsilon\omega_N}{N+1}$$
where $\omega_N$ is the volume of the unit ball in $\Bbb{R}^N$. This would lead to $$ h_\varepsilon=-\eta_\varepsilon \frac{N+1}{\varepsilon \omega_N}$$
However, in the article the answer is 
$$ h_\varepsilon = -\eta_\varepsilon  \frac{N}{ \varepsilon^{\frac{N-1}{N}}\omega_{N-1}}$$
This seems wrong to me. The different constants are not a problem but some facts presented in the following of this rely crucially on the fact that the power of $\varepsilon$ in $h_\varepsilon$ is $\frac{1-N}{N}$ and not $-1$ as I got. 
Is my calculation of the integral $(I)$ correct, or I missed something?

Independent of the details of your calculation, the book's answer can't be right since $|B_\epsilon|$ clearly goes as $\epsilon\omega_N$ and not as $\epsilon^{(N-1)/N}\omega_{N-1}$. It looks as if they were calculating the integral over the sphere rather than the ball, but I don't see why they would do that. Are you sure that $\Omega\subset\mathbb R^n$? Sometimes $\Omega$ is used to denote solid angle.
