Answer
$$0.410$$
Work Step by Step
Given $$ f(x)=f(x)=\cos ^{2} x, \quad\left[\frac{\pi}{6}, \frac{\pi}{2}\right]$$
Since $n=4$, $\Delta x= \dfrac{b-a}{n}=\dfrac{\pi}{12}$ and
$$x_0= \pi/6,\ x_1= \pi/4,\ x_2= \pi/3,\ x_3=5\pi/12,\ x_4= \pi/2$$
Then
\begin{align*}
L_{n}&=\left[f(x_0)+f(x_1)+.......+f(x_{n-1})\right]\Delta x\\
L_4&=\left[f(x_0)+f(x_1)+.......+f(x_{5})\right]\Delta x\\
&=\left[ f(\pi/6)+ f( \pi/4)+ f(\pi /3)+f( 5\pi/12) \right]\frac{\pi}{12}\\
&= \left[ \cos ^{2}(\pi/6)+ \cos ^{2}( \pi/4)+ \cos ^{2}(\pi /3)+\cos ^{2}( 5\pi/12) \right]\frac{\pi}{12}\\
&=\approx 0.410
\end{align*}
