How to calculate the limits of the integral and the outward vector? How to make change of variables in this proof?

Question

I would like to see the calculations behind this proof I found here: Proof of polar coordinates theorem in Evans' PDE Book That is, I am struggling to understand the following:

1. How do we calculate the limits of the integral and the outward normal vector in the proof below?

2. How can I make a change of variables from $\partial B(x_0,r)$ or $B(x_0, r)$ in order to obtain $\partial B(0,1)$ or $B(a+x_0t, r/2)$, with $a, b \in \mathbb{R}$? I mean, how can I calculate the limits of the integral, the outward normal vector, and the jacobian in this case?

I appreciate if you can help.