calculate-A-n-1-n-n-2-e-x-2-3y-2-x-2-3y-2-dxdy-and-find-lim-n-A-n-

Question Number 59169 by maxmathsup by imad last updated on 05/May/19

c a l c u l a t e A_{n} = \int \int_{[\frac{1}{n}, n [^{2}} e^{- x^{2} - 3 y^{2}} \sqrt{x^{2} + 3 y^{2}} d x d y

a n d f i n d {l i m}_{n \to + \infty} A_{n}

Commented by maxmathsup by imad last updated on 07/May/19

l e t c o n s i d e r t h e d i f f e o m o r p h i s m x = r c o s θ a n d y = \frac{1}{\sqrt{3}} r s i n θ

M_{j} = (\begin{matrix} \frac{\partial φ_{1}}{\partial r} \frac{\partial φ_{1}}{\partial θ} \\ \frac{\partial φ_{2}}{\partial r} \frac{\partial φ_{2}}{\partial θ} \end{matrix}) = (\begin{matrix} c o s θ - r s i n θ \\ \frac{s i n θ}{\sqrt{3}} \frac{r}{\sqrt{3}} c o s θ \end{matrix}) \Rightarrow d e t (M_{j}) = \frac{r}{\sqrt{3}}

w e h a v e \frac{1}{n} ⩽ x ⩽ n a n d \frac{1}{n} ⩽ y ⩽ n \Rightarrow \frac{2}{n^{2}} ⩽ x^{2} + y^{2} ⩽ 2 n^{2} \Rightarrow \frac{\sqrt{2}}{n} ⩽ r ⩽ n \sqrt{2} \Rightarrow

A_{n} = \int \int_{\frac{\sqrt{2}}{n} ⩽ r ⩽ n \sqrt{2} a n d 0 ⩽ θ ⩽ \frac{π}{2}} e^{- r^{2}} \sqrt{r^{2}} \frac{r}{\sqrt{3}} d r d θ = \frac{1}{\sqrt{3}} \int_{\frac{\sqrt{2}}{n}}^{n \sqrt{2}} r^{2} e^{- r^{2}} d r . \int_{0}^{\frac{π}{2}} d θ

= \frac{π}{2 \sqrt{3}} \int_{\frac{\sqrt{2}}{n}}^{n \sqrt{2}} r^{2} e^{- r^{2}} d r b y p a r t s u^{^{'}} = r e^{- r^{2}} a n d v = r

\int_{\frac{\sqrt{2}}{n}}^{n \sqrt{2}} r^{2} e^{- r^{2}} d r = {[- \frac{1}{2} r e^{- r^{2}}]}_{\frac{\sqrt{2}}{n}}^{n \sqrt{2}} - \int_{\frac{\sqrt{2}}{n}}^{n \sqrt{2}} - \frac{1}{2} e^{- r^{2}} d r

= - \frac{1}{2} {n \sqrt{2} e^{- 2 n^{2}} - \frac{\sqrt{2}}{n} e^{- \frac{2}{n^{2}}}} + \frac{1}{2} \int_{\frac{\sqrt{2}}{n}}^{n \sqrt{2}} e^{- r^{2}} d r \Rightarrow

A_{n} = - \frac{π}{4 \sqrt{3}} {n \sqrt{2} e^{- 2 n^{2}} - \frac{\sqrt{2}}{n} e^{- \frac{2}{n^{2}}}} + \frac{π}{4 \sqrt{3}} \int_{\frac{\sqrt{2}}{n}}^{n \sqrt{2}} e^{- r^{2}} d r \Rightarrow

{l i m}_{n \to + \infty} A_{n} = \frac{π}{4 \sqrt{3}} \int_{0}^{\infty} e^{- r^{2}} d r = \frac{π}{4 \sqrt{3}} \frac{\sqrt{π}}{2} = \frac{π \sqrt{π}}{8 \sqrt{3}} .