How come I get a negative answer?
What's wrong?
10x in advance.
The variable is r:
Your integrand is $\displaystyle r \sin r$ and you're integrating between $\displaystyle \pi \leq r \leq 2 \pi$. This integrand is always less than or equal to zero for $\displaystyle \pi \leq r \leq 2 \pi$. So it's unsurprising that the result of the integration is less than zero.
New questions require new threads.
You need to switch to a new set of coordinates. The integrand suggests the following transformation:
$\displaystyle u = x - y$ .... (1)
$\displaystyle v = x + y$ .... (2)
(1) and (2) imply $\displaystyle x = \frac{u + v}{2}$ and $\displaystyle y = \frac{v - u}{2}$.
Jacobian of the transformation: $\displaystyle J(u, v) = \frac{1}{2}$.
The region of integration in the xy-plane transforms into the region in the uv-plane bounded by the lines $\displaystyle v = u, ~ v = -u, ~ v = 1$ and $\displaystyle v = 2$.
So the integral becomes $\displaystyle \frac{1}{2} \int_{v=1}^{v=2} \int_{u=-v}^{u=v} e^{u/v} \, du \, dv$.