In David Bachman's "A Geometric Approach to Differential Forms", in example 52, he lists what integrating a particular 1-form is like on $S^1$. My question is, how does he calculate the ranges for the integrals that he does?
Example 52
Let $S^1$, $U_i$, $\phi_i$, and $w$ be defined as in Examples 49 and 50. A partition of unity subordinate to the cover ${\phi_i(U_i)}$ is as follows.
\begin{align*} f_1(x,y) = y^2 \text{ if } y \geq 0 & \text{ else } 0 \\ f_2(x,y) = y^2 \text{ if } y < 0 & \text{ else } 0 \\ f_3(x,y) = x^2 \text{ if } x \geq 0 & \text{ else } 0 \\ f_4(x,y) = x^2 \text{ if } x < 0 & \text{ else } 0. \end{align*} (Check this!) Let $\mu: [0, \pi] \rightarrow S^1$ be defined by $\mu(\theta) = (\cos(\theta), \sin(\theta))$. Then the image of $\mu$ is a 1-cell, $\sigma$, in $S^1$. Let's integrate $w$ over $\sigma$.
$$ \int_\sigma \omega = \sum_{i=1}^4 \int_{\phi^{-1}(\sigma)} \phi_i^*(f_iw). $$
In the next step he shows,
$$ \int_{-(-1,1)} -\sqrt{1-t^2}dt + 0 + \int_{[0,1)} \sqrt{1-t^2}dt + \int_{-[0,1)} -\sqrt{1-t^2}dt. $$
In general, I do not understand how he gets the range of the four integrals. I would expect it to all be $(-1,1)$. Since $\sigma$ is the entire $S^1$ image and the partitions of unity would limit the integrand to be valid only for the points in $S^1$ that each $\phi_i$ outputs to, I don't see why it wouldn't always be $(-1,1)$.
References
$$ \omega = -y\ dx + x\ dy $$
Let $U_i = (-1,1)$ for $i=1,2,3,4$. \begin{align*} \phi_1(t) ={} & (t, \sqrt{1-t^2}) \\ \phi_2(t) ={} & (t, -\sqrt{1-t^2}) \\ \phi_3(t) ={} & (\sqrt{1-t^2}, t) \\ \phi_4(t) ={} & (-\sqrt{1-t^2},t). \end{align*}
\begin{align*} \phi_1^* \omega = \phi_4^* \omega ={} & \frac{-1}{\sqrt{1-t^2}}dt \\ \phi_2^* \omega =\phi_3^* \omega ={} & \frac{-1}{\sqrt{1-t^2}}dt. \end{align*}