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Question Number 59172 by maxmathsup by imad last updated on 05/May/19
calculate ∫∫_([1,3]^2 )     (x+y)ln(x^2  +y^2 )dxdy
$${calculate}\:\int\int_{\left[\mathrm{1},\mathrm{3}\right]^{\mathrm{2}} } \:\:\:\:\left({x}+{y}\right){ln}\left({x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \right){dxdy}\: \\ $$
Commented by maxmathsup by imad last updated on 06/May/19
let use the diffeomorphism  x =rcosθ and y =rsinθ  we have   1≤x≤3 and 1≤y≤3 ⇒2 ≤x^2  +y^2 ≤18 ⇒2≤r^2 ≤18 ⇒(√2)≤r≤3(√2)   ⇒ I =∫∫_((√2)≤r≤3(√2)    and 0≤θ≤(π/2))     r(cosθ +sinθ)(2ln(r)rdr dθ  =2∫_(√2) ^(3(√2))   r^2 ln(r)dr ∫_0 ^(π/2)  (cosθ +sinθ)dθ  by parts  ∫_(√2) ^(3(√2))   r^2 ln(r)dr =[(r^3 /3)ln(r)]_(√2) ^(3(√2))  −∫_(√2) ^(3(√2)) (r^2 /3) dr  =(1/3)(54(√2)ln(3(√2))−((2(√2))/3)ln((√2)) −(1/9)[r^3 ]_(√2) ^(3(√2))   =18(√2)ln(3(√2))−((2(√2))/9)ln((√2))−(1/9)(52(√2))  ∫_0 ^(π/2) (cosθ +sinθ)dθ =[sinθ −cosθ]_0 ^(π/2)  =1+1 =2 ⇒  I =72(√2)ln(3(√2))−((8(√2))/9)ln((√2))−((208)/9) (√2) .
$${let}\:{use}\:{the}\:{diffeomorphism}\:\:{x}\:={rcos}\theta\:{and}\:{y}\:={rsin}\theta\:\:{we}\:{have}\: \\ $$$$\mathrm{1}\leqslant{x}\leqslant\mathrm{3}\:{and}\:\mathrm{1}\leqslant{y}\leqslant\mathrm{3}\:\Rightarrow\mathrm{2}\:\leqslant{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \leqslant\mathrm{18}\:\Rightarrow\mathrm{2}\leqslant{r}^{\mathrm{2}} \leqslant\mathrm{18}\:\Rightarrow\sqrt{\mathrm{2}}\leqslant{r}\leqslant\mathrm{3}\sqrt{\mathrm{2}} \\ $$$$\:\Rightarrow\:{I}\:=\int\int_{\sqrt{\mathrm{2}}\leqslant{r}\leqslant\mathrm{3}\sqrt{\mathrm{2}}\:\:\:\:{and}\:\mathrm{0}\leqslant\theta\leqslant\frac{\pi}{\mathrm{2}}} \:\:\:\:{r}\left({cos}\theta\:+{sin}\theta\right)\left(\mathrm{2}{ln}\left({r}\right){rdr}\:{d}\theta\right. \\ $$$$=\mathrm{2}\int_{\sqrt{\mathrm{2}}} ^{\mathrm{3}\sqrt{\mathrm{2}}} \:\:{r}^{\mathrm{2}} {ln}\left({r}\right){dr}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\left({cos}\theta\:+{sin}\theta\right){d}\theta \\ $$$${by}\:{parts}\:\:\int_{\sqrt{\mathrm{2}}} ^{\mathrm{3}\sqrt{\mathrm{2}}} \:\:{r}^{\mathrm{2}} {ln}\left({r}\right){dr}\:=\left[\frac{{r}^{\mathrm{3}} }{\mathrm{3}}{ln}\left({r}\right)\right]_{\sqrt{\mathrm{2}}} ^{\mathrm{3}\sqrt{\mathrm{2}}} \:−\int_{\sqrt{\mathrm{2}}} ^{\mathrm{3}\sqrt{\mathrm{2}}} \frac{{r}^{\mathrm{2}} }{\mathrm{3}}\:{dr} \\ $$$$=\frac{\mathrm{1}}{\mathrm{3}}\left(\mathrm{54}\sqrt{\mathrm{2}}{ln}\left(\mathrm{3}\sqrt{\mathrm{2}}\right)−\frac{\mathrm{2}\sqrt{\mathrm{2}}}{\mathrm{3}}{ln}\left(\sqrt{\mathrm{2}}\right)\:−\frac{\mathrm{1}}{\mathrm{9}}\left[{r}^{\mathrm{3}} \right]_{\sqrt{\mathrm{2}}} ^{\mathrm{3}\sqrt{\mathrm{2}}} \right. \\ $$$$=\mathrm{18}\sqrt{\mathrm{2}}{ln}\left(\mathrm{3}\sqrt{\mathrm{2}}\right)−\frac{\mathrm{2}\sqrt{\mathrm{2}}}{\mathrm{9}}{ln}\left(\sqrt{\mathrm{2}}\right)−\frac{\mathrm{1}}{\mathrm{9}}\left(\mathrm{52}\sqrt{\mathrm{2}}\right) \\ $$$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left({cos}\theta\:+{sin}\theta\right){d}\theta\:=\left[{sin}\theta\:−{cos}\theta\right]_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:=\mathrm{1}+\mathrm{1}\:=\mathrm{2}\:\Rightarrow \\ $$$${I}\:=\mathrm{72}\sqrt{\mathrm{2}}{ln}\left(\mathrm{3}\sqrt{\mathrm{2}}\right)−\frac{\mathrm{8}\sqrt{\mathrm{2}}}{\mathrm{9}}{ln}\left(\sqrt{\mathrm{2}}\right)−\frac{\mathrm{208}}{\mathrm{9}}\:\sqrt{\mathrm{2}}\:. \\ $$

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