Question Number 59172 by maxmathsup by imad last updated on 05/May/19
$${calculate}\:\int\int_{\left[\mathrm{1},\mathrm{3}\right]^{\mathrm{2}} } \:\:\:\:\left({x}+{y}\right){ln}\left({x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \right){dxdy}\: \\ $$
Commented by maxmathsup by imad last updated on 06/May/19
$${let}\:{use}\:{the}\:{diffeomorphism}\:\:{x}\:={rcos}\theta\:{and}\:{y}\:={rsin}\theta\:\:{we}\:{have}\: \\ $$$$\mathrm{1}\leqslant{x}\leqslant\mathrm{3}\:{and}\:\mathrm{1}\leqslant{y}\leqslant\mathrm{3}\:\Rightarrow\mathrm{2}\:\leqslant{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \leqslant\mathrm{18}\:\Rightarrow\mathrm{2}\leqslant{r}^{\mathrm{2}} \leqslant\mathrm{18}\:\Rightarrow\sqrt{\mathrm{2}}\leqslant{r}\leqslant\mathrm{3}\sqrt{\mathrm{2}} \\ $$$$\:\Rightarrow\:{I}\:=\int\int_{\sqrt{\mathrm{2}}\leqslant{r}\leqslant\mathrm{3}\sqrt{\mathrm{2}}\:\:\:\:{and}\:\mathrm{0}\leqslant\theta\leqslant\frac{\pi}{\mathrm{2}}} \:\:\:\:{r}\left({cos}\theta\:+{sin}\theta\right)\left(\mathrm{2}{ln}\left({r}\right){rdr}\:{d}\theta\right. \\ $$$$=\mathrm{2}\int_{\sqrt{\mathrm{2}}} ^{\mathrm{3}\sqrt{\mathrm{2}}} \:\:{r}^{\mathrm{2}} {ln}\left({r}\right){dr}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\left({cos}\theta\:+{sin}\theta\right){d}\theta \\ $$$${by}\:{parts}\:\:\int_{\sqrt{\mathrm{2}}} ^{\mathrm{3}\sqrt{\mathrm{2}}} \:\:{r}^{\mathrm{2}} {ln}\left({r}\right){dr}\:=\left[\frac{{r}^{\mathrm{3}} }{\mathrm{3}}{ln}\left({r}\right)\right]_{\sqrt{\mathrm{2}}} ^{\mathrm{3}\sqrt{\mathrm{2}}} \:−\int_{\sqrt{\mathrm{2}}} ^{\mathrm{3}\sqrt{\mathrm{2}}} \frac{{r}^{\mathrm{2}} }{\mathrm{3}}\:{dr} \\ $$$$=\frac{\mathrm{1}}{\mathrm{3}}\left(\mathrm{54}\sqrt{\mathrm{2}}{ln}\left(\mathrm{3}\sqrt{\mathrm{2}}\right)−\frac{\mathrm{2}\sqrt{\mathrm{2}}}{\mathrm{3}}{ln}\left(\sqrt{\mathrm{2}}\right)\:−\frac{\mathrm{1}}{\mathrm{9}}\left[{r}^{\mathrm{3}} \right]_{\sqrt{\mathrm{2}}} ^{\mathrm{3}\sqrt{\mathrm{2}}} \right. \\ $$$$=\mathrm{18}\sqrt{\mathrm{2}}{ln}\left(\mathrm{3}\sqrt{\mathrm{2}}\right)−\frac{\mathrm{2}\sqrt{\mathrm{2}}}{\mathrm{9}}{ln}\left(\sqrt{\mathrm{2}}\right)−\frac{\mathrm{1}}{\mathrm{9}}\left(\mathrm{52}\sqrt{\mathrm{2}}\right) \\ $$$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left({cos}\theta\:+{sin}\theta\right){d}\theta\:=\left[{sin}\theta\:−{cos}\theta\right]_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:=\mathrm{1}+\mathrm{1}\:=\mathrm{2}\:\Rightarrow \\ $$$${I}\:=\mathrm{72}\sqrt{\mathrm{2}}{ln}\left(\mathrm{3}\sqrt{\mathrm{2}}\right)−\frac{\mathrm{8}\sqrt{\mathrm{2}}}{\mathrm{9}}{ln}\left(\sqrt{\mathrm{2}}\right)−\frac{\mathrm{208}}{\mathrm{9}}\:\sqrt{\mathrm{2}}\:. \\ $$