I'm trying to figure out if I managed to do the boundary correct in this exercise, no need to evaluate the integral, just let me know if I got them wrong and how should I do them instead.
$$\int \int _D\:\frac{x}{x^2+y^2}dxdy$$ where D: $x^2+y^2\ge 1$ , $0\le y\le x\le 1$
What I have done so far:
$x^2+y^2=1$ circle with the center in $O(0,0)$ and R=$\frac{1}{2}$
$y=0$
$y=x$
$x=1$
$T\rightarrow D':D,$ T: $x=rcos\theta $ , $y=rsin\theta $
So the upper and lower boundary for $\theta$ is $\frac{\pi }{4}\le \theta \le \frac{\pi }{2}$ and the one for r is $\frac{1}{2}\le r\le 1$
Is it correct?
Please see the shaded region that we need to integrate over.
In polar coordinates, $x = r \cos\theta, y = r \sin\theta$
As $x^2 + y^2 \geq 1 \implies r \geq 1$
$r$ is bound above by $x = 1$. $ \ x \leq 1 \implies r \leq \sec\theta$
Now as $x, y \geq 0, \theta \geq 0$. Also, $x \geq y \implies \tan\theta \leq 1, \theta \leq \frac{\pi}{4}$
So bounds are: $1 \leq r \leq \sec\theta$ and $0 \leq \theta \leq \frac{\pi}{4}$. You have to integrate over $dr$ first and then $d\theta$.