Answer
$$0.277 \leq \int_{\pi / 8}^{\pi / 4} \cos x d x \leq 0.363$$
Work Step by Step
Since $\frac{d}{dx}\cos x=-\sin x<0$ on $[\pi/8,\pi/4]$, then $\cos x $ is decreasing and $$f(\pi/8)\leq \cos x\leq f(\pi/4) $$
Hence, by the comparison theorem
\begin{aligned}
m(b-a)& \leq \int_{a}^{b} f(x) d x \leq M(b-a)\\
0.707\left(\frac{\pi}{4}-\frac{\pi}{8}\right) & \leq \int_{\pi / 8}^{\pi / 4} \cos x d x \leq 0.924\left(\frac{\pi}{4}-\frac{\pi}{8}\right) \\
0.707\left(\frac{\pi}{8}\right) & \leq \int_{\pi / 8}^{\pi / 4} \cos x d x \leq 0.924\left(\frac{\pi}{8}\right) \\
0.277 & \leq \int_{\pi / 8}^{\pi / 4} \cos x d x \leq 0.363
\end{aligned}
