The region is a segment of a circle with radius √2 lying above the line [tex]y=1[/tex]. In polar coordinates, this line has equation
[tex]y = r\sin(\theta) = 1 \implies r = \csc(\theta)[/tex]
and the circle has equation
[tex]x^2+y^2 = r^2 = 2 \implies r=\sqrt2[/tex]
The two curves meet when
[tex]\csc(\theta) = \sqrt2 \implies \sin(\theta) = \dfrac1{\sqrt2} \implies \theta = \dfrac\pi4 \text { or } \theta = \dfrac{3\pi}4[/tex]
Then the same integral in polar coordinates is
[tex]\displaystyle \int_{-1}^1 \int_1^{\sqrt{2-x^2}} xy \, dy \, dx = \boxed{\int_{\pi/4}^{3\pi/4} \int_{\csc(\theta)}^{\sqrt2} r^3 \cos(\theta) \sin(\theta) \, dr \, d\theta}[/tex]
