$${calculate}\:\int\int_{{w}} \:\:\:\left({x}^{\mathrm{2}} −\mathrm{2}{y}^{\mathrm{2}} \right)\sqrt{{x}^{\mathrm{2}} +\mathrm{3}{y}^{\mathrm{2}} }{dxdy} \\$$$${with}\:{w}\:=\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} /\:\mathrm{0}\leqslant{x}\leqslant\mathrm{1}\:\:{and}\:\mathrm{1}\leqslant{y}\leqslant\mathrm{2}\right\} \\$$
$${we}\:{use}\:{the}\:{diffeomorphism}\:\:\left({r},\theta\right)\rightarrow\varphi\left({r},\theta\right)=\left({x},{y}\right)=\left({rcos}\theta,\frac{{r}}{\:\sqrt{\mathrm{3}}}{sin}\theta\right) \\$$$$=\left(\varphi_{\mathrm{1}} ,\varphi_{\mathrm{2}} \right)\:\:{we}\:{have}\:\:\mathrm{0}\leqslant{x}^{\mathrm{2}} \:\leqslant\mathrm{1}\:\:{and}\:\mathrm{1}\leqslant{y}^{\mathrm{2}} \leqslant\mathrm{4}\:\Rightarrow\mathrm{3}\leqslant\mathrm{3}{y}^{\mathrm{2}} \:\leqslant\mathrm{12}\:\Rightarrow \\$$$$\mathrm{3}\leqslant{x}^{\mathrm{2}} \:+\mathrm{3}{y}^{\mathrm{2}} \leqslant\mathrm{13}\:\Rightarrow\sqrt{\mathrm{3}}\leqslant\sqrt{{x}^{\mathrm{2}} \:+\mathrm{3}{y}^{\mathrm{2}} }\leqslant\sqrt{\mathrm{13}}\:\Rightarrow\sqrt{\mathrm{3}}\leqslant{r}\leqslant\sqrt{\mathrm{13}} \\$$$${M}_{{j}} \left(\varphi\right)\:=\begin{pmatrix}{\frac{\partial\varphi_{\mathrm{1}} }{\partial{r}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\partial\varphi_{\mathrm{1}} }{\partial\theta}}\\{\frac{\partial\varphi_{\mathrm{2}} }{\partial{r}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\partial\varphi_{\mathrm{2}} }{\partial\theta}\:\:\:\:\:\:}\end{pmatrix} \\$$$$=\begin{pmatrix}{{cos}\theta\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:−{rsin}\theta}\\{\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}}{sin}\theta\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{{r}}{\:\sqrt{\mathrm{3}}}{cos}\theta}\end{pmatrix}\:\:\:\:\:\:\Rightarrow{det}\left({M}_{{j}} \left(\varphi\right)\right)=\frac{{r}}{\:\sqrt{\mathrm{3}}}{cos}^{\mathrm{2}} \theta+\frac{{r}}{\:\sqrt{\mathrm{3}}}{sin}^{\mathrm{2}} \theta \\$$$$=\frac{{r}}{\:\sqrt{\mathrm{3}}}\:\Rightarrow\int\int_{{w}} \left({x}^{\mathrm{2}} −\mathrm{2}{y}^{\mathrm{2}} \right)\sqrt{{x}^{\mathrm{2}} \:+\mathrm{3}{y}^{\mathrm{2}} }{dxdy} \\$$$$=\int\int_{\sqrt{\mathrm{3}}\leqslant{r}\leqslant\sqrt{\mathrm{13}}{and}\:\:\mathrm{0}\leqslant\theta\leqslant\frac{\pi}{\mathrm{2}}} \left({r}^{\mathrm{2}} {cos}^{\mathrm{2}} \theta−\frac{\mathrm{2}}{\mathrm{3}}{r}^{\mathrm{2}} {sin}^{\mathrm{2}} \theta\right){r}\frac{{r}}{\:\sqrt{\mathrm{3}}}{drd}\theta \\$$$$=\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}}\int_{\sqrt{\mathrm{3}}} ^{\sqrt{\mathrm{13}}} {r}^{\mathrm{4}} {dr}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left(\:{cos}^{\mathrm{2}} \theta−\frac{\mathrm{2}}{\mathrm{3}}{sin}^{\mathrm{2}} \theta\right){d}\theta\:\:\:\:\:{we}\:{have} \\$$$$\int_{\sqrt{\mathrm{3}}} ^{\sqrt{\mathrm{13}}} \:{r}^{\mathrm{4}} \:{dr}\:\:=\left[\frac{{r}^{\mathrm{5}} }{\mathrm{5}}\right]_{\sqrt{\mathrm{3}}} ^{\sqrt{\mathrm{13}}} =\frac{\mathrm{1}}{\mathrm{5}}\left\{\left(\sqrt{\mathrm{13}}\right)^{\mathrm{5}} −\left(\sqrt{\mathrm{3}}\right)^{\mathrm{5}} \right. \\$$$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left({cos}^{\mathrm{2}} \theta−\frac{\mathrm{2}}{\mathrm{3}}{sin}^{\mathrm{2}} \theta\right){d}\theta\:=\frac{\mathrm{1}}{\mathrm{3}}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left(\mathrm{3}{cos}^{\mathrm{2}} \theta−\mathrm{2}{sin}^{\mathrm{2}} \theta\right){d}\theta \\$$$$=\frac{\mathrm{1}}{\mathrm{3}}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left(\mathrm{3}\frac{\mathrm{1}+{cos}\left(\mathrm{2}\theta\right)}{\mathrm{2}}−\mathrm{2}\frac{\mathrm{1}−{cos}\left(\mathrm{2}\theta\right)}{\mathrm{2}}\right){d}\theta \\$$$$=\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left(\mathrm{1}+{cos}\left(\mathrm{2}\theta\right)\right){d}\theta−\frac{\mathrm{1}}{\mathrm{3}}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left(\mathrm{1}−{cos}\left(\mathrm{2}\theta\right)\right){d}\theta \\$$$$=\frac{\pi}{\mathrm{4}}\:+\frac{\mathrm{1}}{\mathrm{4}}\left[{sin}\left(\mathrm{2}\theta\right)\right]_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} −\frac{\pi}{\mathrm{6}}\:+\frac{\mathrm{1}}{\mathrm{6}}\left[{sin}\left(\mathrm{2}\theta\right)\right]_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \\$$$$=\frac{\pi}{\mathrm{12}}\:\Rightarrow \\$$$$\int\int_{{w}} \left({x}^{\mathrm{2}} −\mathrm{2}{y}^{\mathrm{2}} \right)\sqrt{{x}^{\mathrm{2}} +\mathrm{3}{y}^{\mathrm{2}} }{dxdy}\:=\frac{\mathrm{1}}{\mathrm{5}\sqrt{\mathrm{3}}}\left\{\left(\sqrt{\mathrm{13}}\right)^{\mathrm{5}} −\left(\sqrt{\mathrm{3}}\right)^{\mathrm{5}} \right\}\frac{\pi}{\mathrm{12}} \\$$$$=\frac{\pi}{\mathrm{60}\sqrt{\mathrm{3}}}\left\{\:\left(\sqrt{\mathrm{13}}\right)^{\mathrm{5}} −\left(\sqrt{\mathrm{3}}\right)^{\mathrm{5}} \right\}. \\$$$$\\$$