Note: Post has been edited as per comment from GerryMyerson.
Area bound by 2 functions $f1$ and $f2$ can be found using integration of the function $f1-f2$
You must first identify the points where the 2 curves intersect in the are you are interested $0 \le x \le \pi$ in your case. This is what you started doing.
so you need to find values of x that satisfies:
$2 \pi \sin x - \pi - x = 0$
There are different ways you can do this depending on your background and tools available to you.
For example, you could either:
Draw the curves
Use numerical analysis
Expand the function using series
Use a software
Using 1,4 I found the points where the 2 curves intersect to have x values of: x1=0.647 and x2=2.142 (good for 2 decimal points)
Define $D(x) = 2 \sin x - 1 - \frac{x}{\pi}$
$A=ABS(A1) + ABS(A2) + ABS(A3)$
Where:
$A1=\int_{0}^{0.647} D(x)$
$A2=\int_{0.647}^{2.142} D(x)$
$A3=\int_{2.142}^{\pi} D(x)$
The diagram below may help you.
