Answer
$\sqrt 3-\frac{\pi}{3}\approx0.685$
Work Step by Step
Step 1. Using the given figure, we have the function $y=sin(x)$ and the interval $[\frac{\pi}{6},\frac{5\pi}{6}]$; the shaded area is the difference between the area under the function and the rectangle with dimensions $a\times b$.
Step 2. We have $a=\frac{5\pi}{6}-\frac{\pi}{6}=\frac{2\pi}{3}$ and $b=sin(\frac{\pi}{6})=\frac{1}{2}$; thus the area of the rectangle is $A_2=\frac{\pi}{3}$
Step 3. For the area under the function, we have $A_1=\int_{\frac{\pi}{6}}^{\frac{5\pi}{6}}(sin(x))dx=(-cos(x))|_{\frac{\pi}{6}}^{\frac{5\pi}{6}}=(-cos(\frac{5\pi}{6}))-(-cos(\frac{\pi}{6}))=\sqrt 3$
Step 4. The shaded area is given by $A=A_1-A_2=\sqrt 3-\frac{\pi}{3}\approx0.685$
