[tex]x=r\cos\phi\\y=r\sin\phi\\\det J=r[/tex]
[tex]x^2+y^2=r^2\leq 4\\0\leq r\leq 2[/tex]
[tex]x\leq0\\r\cos\phi\leq 0\\\cos\phi\leq0\\\frac{\pi}{2}\leq\phi\leq\frac{3\pi}{2}[/tex]
[tex]\int_{\pi/2}^{3\pi/2}{d\phi}\int_{0}^2{r^2\cdot r\,dr}=\pi\cdot\frac{r^4}{4}|_0^2=4\pi[/tex]
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