Area under conformal Mapping $ e^z $

Question

The question is as follows:

Consider the region

$\Omega = {x+iy: -1 \le x \le 2, -\pi/3 \le y \le \pi/3 } $

in the complex plane. The transformation $x + iy → e^{x+iy}$ maps the region Ω onto the region S ⊂ ℂ (the set of all complex numbers). Then the area of the region S is equal to.

I managed to find the relationship between (x,y) in input plane to (u,v) in the output plane as:

$y = tan^-1(v/u)$

$x = 1/2 ln(v^2 + u^2)$

Using this and the conditions in the question, I got the area as the red shaded region shown below

The two circles have radii $e^4$ and $e^{-2}$.

The answer should come out to be $2\pi/3 (e^4 - e^{-2})$

But the answer shown in solution paper is $\pi/3 (e^4 - e^{-2})$

My question is, is it not necessary to consider the left half of U-V plane for this?

Answer 1

$u = e^x cosy \ge 0$

So only the region in the right half plane matters