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Question Number 78286 by msup trace by abdo last updated on 15/Jan/20
$${find}\:{A}_{{n}} =\int\int_{\left[\mathrm{0},{n}\left[\right.\right.} \:\:{e}^{−\left({x}^{\mathrm{2}} +\mathrm{3}{y}^{\mathrm{2}} \right)} {sin}\left({x}^{\mathrm{2}} +\mathrm{3}{y}^{\mathrm{2}} \right){dxdy} \\ $$$${and}\:{lim}_{{n}\rightarrow+\infty} \:{A}_{{n}} \\ $$$${find}\:{nature}\:{of}\:{the}\:{serie}\:\Sigma{n}\:{A}_{{n}} \\ $$
Commented by mathmax by abdo last updated on 16/Jan/20
![changement x=rcosθ and y =(r/(√3))sinθ give x^2 +3y^2 =r^2 0≤x≤n and 0≤y≤n ⇒0≤x^2 +y^2 ≤2n^2 ⇒0≤r≤n(√2) θ ∈[0,(π/2)] diffemorphisme is (r,θ)→(x,y)=(ϕ_1 ,ϕ_2 )(rcosθ,(r/(√3))sinθ) M_j (ϕ) = ((((∂ϕ_1 /∂r) (∂ϕ_1 /∂θ))),(((∂ϕ_2 /∂r) (∂ϕ_2 /∂θ))) ) = (((cosθ −rsinθ )),((((sinθ)/(√3)) (r/(√3))cosθ)) ) ⇒det(M_j ) =(r/(√3)) ⇒ A_n =∫_0 ^(π/2) ∫_0 ^(n(√2)) e^(−r^2 ) sin(r^2 )(r/(√3))dr dθ =(π/(2(√3))) ∫_0 ^(n(√2)) r e^(−r^2 ) sin(r^2 )dr by parts u^′ =re^(−r^2 ) and v=sin(r^2 ) A_n =(π/(2(√3))){ [−(1/2)e^(−r^2 ) sin(r^2 )]_0 ^(n(√2)) −∫_0 ^(n(√2)) (−(1/2)e^(−r^2 ) ×2rcos(r^2 ))dr} =(π/(2(√3))){−(1/2)e^(−2n^2 ) sin(2n^2 )+∫_0 ^(n(√2)) r e^(−r^2 ) cos(r^2 )dr} =(π/(2(√3))){ −(1/2)e^(−2n^2 ) sin(2n^2 )+[−(1/2)e^(−r^2 ) cos(r^2 )]_0 ^(n(√2)) −∫_0 ^(n(√2)) (−(1/2)e^(−r^2 ) (−2r sin(r^2 ))dr =(π/(2(√3))){−(1/2)e^(−2n^2 ) sin(2n^2 )+(1/2)−(1/2)e^(−2n^2 ) cos(2n^2 )−∫_0 ^(n(√2)) r e^(−r^2 ) sin(r^2 )dr} ⇒2A_n =(π/(4(√3))){1−e^(−2n^2 ) sin(2n^2 )−e^(−2n^2 ) cos(2n^2 )} ⇒ A_n =(π/(8(√3))){ 1−e^(−2n^2 ) sin(2n^2 )−e^(−2n^2 ) cos(2n^2 )} we have lim_(n→+∞) e^(−2n^2 ) sin(2n^2 )=lim_(n→+∞) e^(−2n^2 ) cos(2n^2 )=0 ⇒ lim_(n→+∞) A_n =(π/(8(√3)))](Q78388.png)
$${changement}\:{x}={rcos}\theta\:\:{and}\:{y}\:=\frac{{r}}{\sqrt{\mathrm{3}}}{sin}\theta\:\:\:{give}\:{x}^{\mathrm{2}} +\mathrm{3}{y}^{\mathrm{2}} ={r}^{\mathrm{2}} \\ $$$$\mathrm{0}\leqslant{x}\leqslant{n}\:{and}\:\mathrm{0}\leqslant{y}\leqslant{n}\:\Rightarrow\mathrm{0}\leqslant{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \leqslant\mathrm{2}{n}^{\mathrm{2}} \:\Rightarrow\mathrm{0}\leqslant{r}\leqslant{n}\sqrt{\mathrm{2}} \\ $$$$\theta\:\in\left[\mathrm{0},\frac{\pi}{\mathrm{2}}\right]\:\:\:{diffemorphisme}\:{is}\:\left({r},\theta\right)\rightarrow\left({x},{y}\right)=\left(\varphi_{\mathrm{1}} ,\varphi_{\mathrm{2}} \right)\left({rcos}\theta,\frac{{r}}{\sqrt{\mathrm{3}}}{sin}\theta\right) \\ $$$${M}_{{j}} \left(\varphi\right)\:=\begin{pmatrix}{\frac{\partial\varphi_{\mathrm{1}} }{\partial{r}}\:\:\:\:\:\:\:\:\:\:\:\frac{\partial\varphi_{\mathrm{1}} }{\partial\theta}}\\{\frac{\partial\varphi_{\mathrm{2}} }{\partial{r}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\partial\varphi_{\mathrm{2}} }{\partial\theta}}\end{pmatrix}\:\:\:\:=\begin{pmatrix}{{cos}\theta\:\:\:\:\:\:\:\:\:\:\:\:−{rsin}\theta\:\:\:\:}\\{\frac{{sin}\theta}{\sqrt{\mathrm{3}}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{{r}}{\sqrt{\mathrm{3}}}{cos}\theta}\end{pmatrix} \\ $$$$\Rightarrow{det}\left({M}_{{j}} \right)\:=\frac{{r}}{\sqrt{\mathrm{3}}}\:\Rightarrow\:{A}_{{n}} =\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \int_{\mathrm{0}} ^{{n}\sqrt{\mathrm{2}}} \:{e}^{−{r}^{\mathrm{2}} } \:{sin}\left({r}^{\mathrm{2}} \right)\frac{{r}}{\sqrt{\mathrm{3}}}{dr}\:{d}\theta \\ $$$$=\frac{\pi}{\mathrm{2}\sqrt{\mathrm{3}}}\:\int_{\mathrm{0}} ^{{n}\sqrt{\mathrm{2}}} \:{r}\:{e}^{−{r}^{\mathrm{2}} } \:{sin}\left({r}^{\mathrm{2}} \right){dr}\:\:{by}\:{parts}\:\:{u}^{'} ={re}^{−{r}^{\mathrm{2}} } \:{and}\:{v}={sin}\left({r}^{\mathrm{2}} \right) \\ $$$${A}_{{n}} =\frac{\pi}{\mathrm{2}\sqrt{\mathrm{3}}}\left\{\:\:\left[−\frac{\mathrm{1}}{\mathrm{2}}{e}^{−{r}^{\mathrm{2}} } {sin}\left({r}^{\mathrm{2}} \right)\right]_{\mathrm{0}} ^{{n}\sqrt{\mathrm{2}}} −\int_{\mathrm{0}} ^{{n}\sqrt{\mathrm{2}}} \left(−\frac{\mathrm{1}}{\mathrm{2}}{e}^{−{r}^{\mathrm{2}} } ×\mathrm{2}{rcos}\left({r}^{\mathrm{2}} \right)\right){dr}\right\} \\ $$$$=\frac{\pi}{\mathrm{2}\sqrt{\mathrm{3}}}\left\{−\frac{\mathrm{1}}{\mathrm{2}}{e}^{−\mathrm{2}{n}^{\mathrm{2}} } {sin}\left(\mathrm{2}{n}^{\mathrm{2}} \right)+\int_{\mathrm{0}} ^{{n}\sqrt{\mathrm{2}}} \:\:{r}\:{e}^{−{r}^{\mathrm{2}} } {cos}\left({r}^{\mathrm{2}} \right){dr}\right\} \\ $$$$=\frac{\pi}{\mathrm{2}\sqrt{\mathrm{3}}}\left\{\:\:−\frac{\mathrm{1}}{\mathrm{2}}{e}^{−\mathrm{2}{n}^{\mathrm{2}} } {sin}\left(\mathrm{2}{n}^{\mathrm{2}} \right)+\left[−\frac{\mathrm{1}}{\mathrm{2}}{e}^{−{r}^{\mathrm{2}} } {cos}\left({r}^{\mathrm{2}} \right)\right]_{\mathrm{0}} ^{{n}\sqrt{\mathrm{2}}} \right. \\ $$$$−\int_{\mathrm{0}} ^{{n}\sqrt{\mathrm{2}}} \left(−\frac{\mathrm{1}}{\mathrm{2}}{e}^{−{r}^{\mathrm{2}} } \left(−\mathrm{2}{r}\:{sin}\left({r}^{\mathrm{2}} \right)\right){dr}\right. \\ $$$$=\frac{\pi}{\mathrm{2}\sqrt{\mathrm{3}}}\left\{−\frac{\mathrm{1}}{\mathrm{2}}{e}^{−\mathrm{2}{n}^{\mathrm{2}} } {sin}\left(\mathrm{2}{n}^{\mathrm{2}} \right)+\frac{\mathrm{1}}{\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{2}}{e}^{−\mathrm{2}{n}^{\mathrm{2}} } {cos}\left(\mathrm{2}{n}^{\mathrm{2}} \right)−\int_{\mathrm{0}} ^{{n}\sqrt{\mathrm{2}}} {r}\:{e}^{−{r}^{\mathrm{2}} } {sin}\left({r}^{\mathrm{2}} \right){dr}\right\} \\ $$$$\Rightarrow\mathrm{2}{A}_{{n}} =\frac{\pi}{\mathrm{4}\sqrt{\mathrm{3}}}\left\{\mathrm{1}−{e}^{−\mathrm{2}{n}^{\mathrm{2}} } {sin}\left(\mathrm{2}{n}^{\mathrm{2}} \right)−{e}^{−\mathrm{2}{n}^{\mathrm{2}} } {cos}\left(\mathrm{2}{n}^{\mathrm{2}} \right)\right\}\:\Rightarrow \\ $$$${A}_{{n}} =\frac{\pi}{\mathrm{8}\sqrt{\mathrm{3}}}\left\{\:\mathrm{1}−{e}^{−\mathrm{2}{n}^{\mathrm{2}} } {sin}\left(\mathrm{2}{n}^{\mathrm{2}} \right)−{e}^{−\mathrm{2}{n}^{\mathrm{2}} } {cos}\left(\mathrm{2}{n}^{\mathrm{2}} \right)\right\} \\ $$$${we}\:{have}\:{lim}_{{n}\rightarrow+\infty} \:{e}^{−\mathrm{2}{n}^{\mathrm{2}} } {sin}\left(\mathrm{2}{n}^{\mathrm{2}} \right)={lim}_{{n}\rightarrow+\infty} \:{e}^{−\mathrm{2}{n}^{\mathrm{2}} } {cos}\left(\mathrm{2}{n}^{\mathrm{2}} \right)=\mathrm{0}\:\Rightarrow \\ $$$${lim}_{{n}\rightarrow+\infty} \:{A}_{{n}} =\frac{\pi}{\mathrm{8}\sqrt{\mathrm{3}}} \\ $$