<s>[INST] Double Integration in region $A=\{ (x,y)\in \mathbb{R}^{2} : x^{2}+y^{2}\le 1, x+y \ge 1, y \le x\}$. 
Calculate
$$\iint _{A} (x^{2}+y^{2})^{-3/2} \,dx\,dy\,,$$
where $A=\{ (x,y)\in \mathbb{R}^{2} : x^{2}+y^{2}\le 1, x+y \ge 1, y \le x\}$.

I first found the intersection points that are $(1,0)$, $\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)$ and $\left(\frac{1}{2}, \frac{1}{2}\right)$. And the determined the regions
\begin{split}
D & =\left\{(x,y)\in \mathbb{R}^{2}: x^{2}+y^{2}, y \le x\} = \{(r,\theta)\in \mathbb{R}^{2}: 0 \le r \le 1, 0 \le \theta \le \frac{\pi}{4} \right\}\\
B & =\left\{(x,y)\in \mathbb{R}^{2}: 0 \le x \le \frac{1}{\sqrt{2}}, 0 \le y \le x\right\}\\
C & =\left\{(x,y)\in \mathbb{R}^{2}: \frac{1}{\sqrt{2}} \le x \le 1,  \frac{1}{\sqrt{2}} \le y \le 1-x \right\}
\end{split}
Then,
$$\iint _{A} (x^{2}+y^{2})^{-3/2} \,dx\,dy = \iint_{D} (x^{2}+y^{2})^{-3/2} \,dx\,dy - \iint_{B} (x^{2}+y^{2})^{-3/2} \,dx\,dy - \iint_{C} (x^{2}+y^{2})^{-3/2} \,dx\,dy$$
Is this right? Or is there any other easiest way?
 [/INST] You are trying to integrate over the the shaded region in the diagram.

There are couple of ways to simplify this -
i) Use polar coordinates to avoid splitting the integral. In polar coordinates, $x = r \cos\theta, y = r \sin\theta, x^2+y^2 = r^2$
So line $x+y=1$ can be rewritten as $r ({\cos \theta + \sin\theta}) = 1$
leading to bounds $\frac{1}{\cos \theta + \sin\theta} \leq r \leq 1$
Also $y = x$ can be rewritten as $\tan\theta = 1, \theta = \frac{\pi}{4}$ leading to bounds $0 \leq \theta \leq \frac{\pi}{4}$
So the integral becomes,
$ \ \displaystyle \int_0^{\pi/4} \int_{1 / (\cos\theta + \sin\theta)}^1 \frac{1}{r^3} \cdot r \ dr \ d\theta = \int_0^{\pi/4} \int_{1 / (\cos\theta + \sin\theta)}^1 \frac{1}{r^3} \cdot r \ dr \ d\theta$
$\displaystyle  = \int_0^{\pi/4} \big[-\frac{1}{r}\big]_{1 / (\cos\theta + \sin\theta)}^1 \ d\theta$
$\displaystyle  = \int_0^{\pi/4} (\cos\theta + \sin\theta - 1) \ d\theta = 1 - \frac{\pi}{4} $
ii) Rotate the region by $\frac{\pi}{4}$ anti-clockwise (though not necessary here).
$x = r \cos (\theta - \frac{\pi}{4}), y = r \sin (\theta - \frac{\pi}{4}), x^2 + y^2 = r^2$
$x + y = 1$ becomes $\sqrt2 r \sin\theta = 1 \implies r = \frac{1}{\sqrt2 \sin\theta}$ and the integral becomes,
$\displaystyle \int_{\pi/4}^{\pi/2} \int_{\csc\theta / \sqrt2}^1 \frac{1}{r^2} \ dr \ d\theta $
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