Lemma 4.4 in Gilbarg-Trudinger In equation (4.12) of the proof of Lemma 4.4 (2nd inequality, 2nd term), Gilbarg-Trudinger uses following inequality
$$\frac{1}{\omega_n}\int_{B}|x-y|^{\alpha-n}dy\leq \frac{n}{\alpha}(3R)^\alpha,$$
where $B=B_x(2R)$ for some fixed radius $R>0$, $\alpha\in (0,1)$, and $\omega_n=\text{volume of unit ball in }\mathbb{R}^n$. I am confused to why there's a "3" on the RHS, and would appreciate if someone could tell me where the mistake in my computation is. I'm sure this is completely elementary. We compute
\begin{align*}
\int_B |x-y|^{\alpha-n}dy&=\frac{n\omega_n(2R)^{n-1}}{(2R)^{n-1}}\int_{0}^{2R}\rho^{\alpha-n}(\rho^{n-1}d\rho)\\
&=(n\omega_n)\frac{\rho^\alpha}{\alpha}\bigg\vert_0^{2R}\\
&=\frac{n\omega_n}{\alpha}(2R)^\alpha,
\end{align*}
where the first equality uses co-area formula for the ball of radius 2R, along with change of variables. The rest is obvious. This "3" appears later in the proof when estimating $|I_3|$, so I'm fairly sure I'm missing something.

The proof starts with "For any $x$ in $B_1$..", with $B_i = B_i(x_0)$ (as opposed to $B_2 = B_2(x)$, what you seem to be assuming in your calculation).
