Suppose $f: S^1 \to S^1$, and $f(x_i) = y$, where $x_i$s are the preimage of a regular value $y$. Then how can I compute $\int_0^{x_0}f^\prime(x), \int_{x_0}^{x_1}f^\prime(x),$?
I realize that $\operatorname{deg}f|_{[0, x_0]}:= \pm 1$ since there is only 1 preimage point of $y$, namely $x_0$.
I am also aware that $\int_0^{x_0} f^\prime(t)dt = f(x_0) - f(0).$
But then it becomes strange: because $x_i$s are preimages of $y$, then $\int_0^{x_0}f^\prime(x) = f(x_0) - f(0) = y - f(0)$ and
$\int_{x_0}^{x_1}f^\prime(x) = f(x_1) - f(x_0) = y - y =0$?