Edit: Completely rewritten to correct a misreading of the original problem.
You know that the graphs of $y=\tan x$ and $y=2\sin x$ cross at $x=0$. Where else do they cross? You need to solve $2\sin x=\tan x=\frac{\sin x}{\cos x}\;,$ or $2\sin x\cos x=\sin x\;.$ One solution is $\sin x=0$, which in the interval from $-\pi/3$ to $\pi/3$ means that $x=0$; we already knew about that. If $\sin x\ne 0$, we can divide through by it to find that $2\cos x=1$, or $\cos x=1/2$. That happens at $x=\pi/3$ and $x=-\pi/3$, so the curves actually cross at the ends of your interval as well as in the middle.
Now, which one is on top where? A really close look at the graphs with a graphing calculator will probably tell you, but you don’t need that tool. Just look at what happens at $x=\frac{\pi}4$: $2\sin\frac{\pi}4=\sqrt2$, and $\tan\frac{\pi}4=1$, so at $x=\frac{\pi}4$, the $y=2\sin x$ curve is above the $y=\tan x$ curve. We know that the curves don’t cross between $x=0$ and $x=\frac{\pi}3$, so the $y=2\sin x$ curve must be above the $y=\tan x$ curve over the entire interval from $x=0$ to $x=\frac{\pi}3$. This means that the length of the vertical strip at some $x$ between $0$ and $\frac{\pi}3$ is top $y$-coordinate minus bottom $y$-coordinate, or $2\sin x-\tan x$.
For $-\pi/3\le x\le 0$ you can either notice that the strips now run from a low $y$ value of $2\sin x$ up to a high $y$ value of $\tan x$ and therefore have length $\tan x-2\sin x$, or you can use your observation that the functions are odd to say that the lefthand area must be equal to the righthand area. Either way, your final result is just the sum of the left- and righthand areas.
Added: In view of your last comment, I thought that I’d finish the calculation; perhaps you’ll easily be able to track down where you went astray. (If you did: sometimes the answer at the back of the book is wrong, even in a well-established text like Stewart’s.)
$\begin{align*}\int_0^{\pi/3}(2\sin x-\tan x)dx&=\Big[-2\cos x+\ln|\cos x|\Big]_0^{\pi/3}\\ &=-2\cos\frac{\pi}3-(-2\cos 0)+\ln\left|\cos\frac{\pi}3\right|-\ln|\cos 0|\\ &=-1-(-2)+\ln\frac12-\ln 1\\ &=1+\ln\frac12+0\\ &=1+(\ln 1-\ln 2)\\ &=1-\ln 2\;. \end{align*}$
This is the area of the righthand half, so by symmetry the total area is twice this, or $2-2\ln 2$.