As part of an example where $\int_{-\infty}^{+\infty} \frac{\cos x}{x}~dx$ is calculated using Complex Analysis and the Residue theorem, my textbook casually states that where $\mu_R^4(t) = R e^{i t}$ for $o \le t \le \pi$, $$ \mathrm{length}(\mu_R^4 \cap \{ z : y < \sqrt{R} \}) \le 4 \sqrt{R} $$ and $$ \mathrm{length}(\mu_R^4 \cap \{z : y \ge \sqrt{R} \}) \le \pi R. $$
Obviously in the second case, the length of the whole semicircle $\mu_R^4$ is precisely $\pi R$, but I do not see how the first estimate was achieved. Is there something obvious I'm missing?