Question Number 59169 by maxmathsup by imad last updated on 05/May/19
$${calculate}\:{A}_{{n}} =\int\int_{\left[\frac{\mathrm{1}}{{n}},{n}\left[^{\mathrm{2}} \right.\right.} \:\:\:\:{e}^{−{x}^{\mathrm{2}} −\mathrm{3}{y}^{\mathrm{2}} } \sqrt{{x}^{\mathrm{2}} \:+\mathrm{3}{y}^{\mathrm{2}} }{dxdy} \\ $$$${and}\:{find}\:{lim}_{{n}\rightarrow+\infty} \:{A}_{{n}} \\ $$
Commented by maxmathsup by imad last updated on 07/May/19
![let consider the diffeomorphism x=rcosθ and y =(1/(√3))rsinθ M_j = ((((∂ϕ_1 /∂r) (∂ϕ_1 /∂θ))),(((∂ϕ_2 /∂r) (∂ϕ_2 /∂θ))) ) = (((cosθ −rsinθ)),((((sinθ)/(√3)) (r/(√3))cosθ)) ) ⇒det(M_j )=(r/(√3)) we have (1/n) ≤x≤n and (1/n)≤y≤n ⇒(2/n^2 ) ≤x^2 +y^2 ≤2n^2 ⇒((√2)/n) ≤r≤n(√(2 )) ⇒ A_n =∫∫_(((√2)/n)≤r≤n(√2) and 0≤θ≤(π/2)) e^(−r^2 ) (√r^2 )(r/(√3)) dr dθ =(1/(√3)) ∫_((√2)/n) ^(n(√2)) r^2 e^(−r^2 ) dr .∫_0 ^(π/2) dθ =(π/(2(√3))) ∫_((√2)/n) ^(n(√2)) r^2 e^(−r^2 ) dr by parts u^′ =r e^(−r^2 ) and v =r ∫_((√2)/n) ^(n(√2)) r^2 e^(−r^2 ) dr =[−(1/2)r e^(−r^2 ) ]_((√2)/n) ^(n(√2)) −∫_((√2)/n) ^(n(√2)) −(1/2) e^(−r^2 ) dr =−(1/2){n(√2)e^(−2n^2 ) −((√2)/n) e^(−(2/n^2 )) } +(1/2) ∫_(((√2)/n) ) ^(n(√2)) e^(−r^2 ) dr ⇒ A_n =−(π/(4(√3))){ n(√2) e^(−2n^2 ) −((√2)/n) e^(−(2/n^2 )) } +(π/(4(√3))) ∫_(((√2)/n) ) ^(n(√2)) e^(−r^2 ) dr ⇒ lim_(n→+∞) A_n =(π/(4(√3))) ∫_0 ^∞ e^(−r^2 ) dr =(π/(4(√3))) ((√π)/2) =((π(√π))/(8(√3))) .](Q59297.png)
$${let}\:{consider}\:{the}\:{diffeomorphism}\:\:{x}={rcos}\theta\:\:{and}\:{y}\:=\frac{\mathrm{1}}{\sqrt{\mathrm{3}}}{rsin}\theta \\ $$$${M}_{{j}} =\begin{pmatrix}{\frac{\partial\varphi_{\mathrm{1}} }{\partial{r}}\:\:\:\:\:\:\:\:\:\:\frac{\partial\varphi_{\mathrm{1}} }{\partial\theta}}\\{\frac{\partial\varphi_{\mathrm{2}} }{\partial{r}}\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\partial\varphi_{\mathrm{2}} }{\partial\theta}}\end{pmatrix}\:\:\:\:\:\:=\begin{pmatrix}{{cos}\theta\:\:\:\:\:\:\:\:\:\:\:\:\:−{rsin}\theta}\\{\frac{{sin}\theta}{\sqrt{\mathrm{3}}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{{r}}{\sqrt{\mathrm{3}}}{cos}\theta}\end{pmatrix}\:\:\Rightarrow{det}\left({M}_{{j}} \right)=\frac{{r}}{\sqrt{\mathrm{3}}} \\ $$$${we}\:{have}\:\frac{\mathrm{1}}{{n}}\:\leqslant{x}\leqslant{n}\:\:{and}\:\:\frac{\mathrm{1}}{{n}}\leqslant{y}\leqslant{n}\:\Rightarrow\frac{\mathrm{2}}{{n}^{\mathrm{2}} }\:\leqslant{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \:\leqslant\mathrm{2}{n}^{\mathrm{2}} \:\Rightarrow\frac{\sqrt{\mathrm{2}}}{{n}}\:\leqslant{r}\leqslant{n}\sqrt{\mathrm{2}\:}\:\Rightarrow \\ $$$${A}_{{n}} =\int\int_{\frac{\sqrt{\mathrm{2}}}{{n}}\leqslant{r}\leqslant{n}\sqrt{\mathrm{2}}\:\:{and}\:\mathrm{0}\leqslant\theta\leqslant\frac{\pi}{\mathrm{2}}} \:\:\:\:{e}^{−{r}^{\mathrm{2}} } \sqrt{{r}^{\mathrm{2}} }\frac{{r}}{\sqrt{\mathrm{3}}}\:{dr}\:{d}\theta\:=\frac{\mathrm{1}}{\sqrt{\mathrm{3}}}\:\int_{\frac{\sqrt{\mathrm{2}}}{{n}}} ^{{n}\sqrt{\mathrm{2}}} {r}^{\mathrm{2}} \:{e}^{−{r}^{\mathrm{2}} } {dr}\:.\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:{d}\theta \\ $$$$=\frac{\pi}{\mathrm{2}\sqrt{\mathrm{3}}}\:\int_{\frac{\sqrt{\mathrm{2}}}{{n}}} ^{{n}\sqrt{\mathrm{2}}} \:\:{r}^{\mathrm{2}} \:{e}^{−{r}^{\mathrm{2}} } {dr}\:\:\:{by}\:{parts}\:\:{u}^{'} \:={r}\:{e}^{−{r}^{\mathrm{2}} } \:\:\:{and}\:{v}\:={r} \\ $$$$\int_{\frac{\sqrt{\mathrm{2}}}{{n}}} ^{{n}\sqrt{\mathrm{2}}} \:\:{r}^{\mathrm{2}} \:{e}^{−{r}^{\mathrm{2}} } {dr}\:=\left[−\frac{\mathrm{1}}{\mathrm{2}}{r}\:{e}^{−{r}^{\mathrm{2}} } \:\:\right]_{\frac{\sqrt{\mathrm{2}}}{{n}}} ^{{n}\sqrt{\mathrm{2}}} \:\:\:\:−\int_{\frac{\sqrt{\mathrm{2}}}{{n}}} ^{{n}\sqrt{\mathrm{2}}} \:\:−\frac{\mathrm{1}}{\mathrm{2}}\:{e}^{−{r}^{\mathrm{2}} } {dr} \\ $$$$=−\frac{\mathrm{1}}{\mathrm{2}}\left\{{n}\sqrt{\mathrm{2}}{e}^{−\mathrm{2}{n}^{\mathrm{2}} } \:\:−\frac{\sqrt{\mathrm{2}}}{{n}}\:{e}^{−\frac{\mathrm{2}}{{n}^{\mathrm{2}} }} \:\:\:\:\:\:\right\}\:\:\:+\frac{\mathrm{1}}{\mathrm{2}}\:\int_{\frac{\sqrt{\mathrm{2}}}{{n}}\:} ^{{n}\sqrt{\mathrm{2}}} \:\:\:{e}^{−{r}^{\mathrm{2}} } {dr}\:\Rightarrow \\ $$$${A}_{{n}} =−\frac{\pi}{\mathrm{4}\sqrt{\mathrm{3}}}\left\{\:\:{n}\sqrt{\mathrm{2}}\:{e}^{−\mathrm{2}{n}^{\mathrm{2}} } \:\:\:\:−\frac{\sqrt{\mathrm{2}}}{{n}}\:{e}^{−\frac{\mathrm{2}}{{n}^{\mathrm{2}} }} \right\}\:+\frac{\pi}{\mathrm{4}\sqrt{\mathrm{3}}}\:\int_{\frac{\sqrt{\mathrm{2}}}{{n}}\:} ^{{n}\sqrt{\mathrm{2}}} \:\:\:\:{e}^{−{r}^{\mathrm{2}} } {dr}\:\:\Rightarrow \\ $$$${lim}_{{n}\rightarrow+\infty} \:{A}_{{n}} =\frac{\pi}{\mathrm{4}\sqrt{\mathrm{3}}}\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{r}^{\mathrm{2}} } {dr}\:=\frac{\pi}{\mathrm{4}\sqrt{\mathrm{3}}}\:\frac{\sqrt{\pi}}{\mathrm{2}}\:=\frac{\pi\sqrt{\pi}}{\mathrm{8}\sqrt{\mathrm{3}}}\:. \\ $$