find A_n =∫∫_([0,n[) e^(−(x^2 +3y^2 )) sin(x^2 +3y^2 )dxdy and lim_(n→+∞) A_n find nature of the serie Σn A


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Question Number 78286 by msup trace by abdo last updated on 15/Jan/20

$f i n d A_{n} = \int \int_{[0, n [} e^{- (x^{2} + 3 y^{2})} s i n (x^{2} + 3 y^{2}) d x d y$ $a n d {l i m}_{n \to + \infty} A_{n}$ $f i n d n a t u r e o f t h e s e r i e Σ n A_{n}$
Commented by mathmax by abdo last updated on 16/Jan/20
$changement x=rcosθ and y =(r/(√3))sinθ give x^2 +3y^2 =r^2 0≤x≤n and 0≤y≤n ⇒0≤x^2 +y^2 ≤2n^2 ⇒0≤r≤n(√2) θ ∈[0,(π/2)] diffemorphisme is (r,θ)→(x,y)=(ϕ_1 ,ϕ_2 )(rcosθ,(r/(√3))sinθ) M_j (ϕ) = ((((∂ϕ_1 /∂r) (∂ϕ_1 /∂θ))),(((∂ϕ_2 /∂r) (∂ϕ_2 /∂θ))) ) = (((cosθ −rsinθ )),((((sinθ)/(√3)) (r/(√3))cosθ)) ) ⇒det(M_j ) =(r/(√3)) ⇒ A_n =∫_0 ^(π/2) ∫_0 ^(n(√2)) e^(−r^2 ) sin(r^2 )(r/(√3))dr dθ =(π/(2(√3))) ∫_0 ^(n(√2)) r e^(−r^2 ) sin(r^2 )dr by parts u^′ =re^(−r^2 ) and v=sin(r^2 ) A_n =(π/(2(√3))){ [−(1/2)e^(−r^2 ) sin(r^2 )]_0 ^(n(√2)) −∫_0 ^(n(√2)) (−(1/2)e^(−r^2 ) ×2rcos(r^2 ))dr} =(π/(2(√3))){−(1/2)e^(−2n^2 ) sin(2n^2 )+∫_0 ^(n(√2)) r e^(−r^2 ) cos(r^2 )dr} =(π/(2(√3))){ −(1/2)e^(−2n^2 ) sin(2n^2 )+[−(1/2)e^(−r^2 ) cos(r^2 )]_0 ^(n(√2)) −∫_0 ^(n(√2)) (−(1/2)e^(−r^2 ) (−2r sin(r^2 ))dr =(π/(2(√3))){−(1/2)e^(−2n^2 ) sin(2n^2 )+(1/2)−(1/2)e^(−2n^2 ) cos(2n^2 )−∫_0 ^(n(√2)) r e^(−r^2 ) sin(r^2 )dr} ⇒2A_n =(π/(4(√3))){1−e^(−2n^2 ) sin(2n^2 )−e^(−2n^2 ) cos(2n^2 )} ⇒ A_n =(π/(8(√3))){ 1−e^(−2n^2 ) sin(2n^2 )−e^(−2n^2 ) cos(2n^2 )} we have lim_(n→+∞) e^(−2n^2 ) sin(2n^2 )=lim_(n→+∞) e^(−2n^2 ) cos(2n^2 )=0 ⇒ lim_(n→+∞) A_n =(π/(8(√3)))$
$c h a n g e m e n t x = r c o s θ a n d y = \frac{r}{\sqrt{3}} s i n θ g i v e x^{2} + 3 y^{2} = r^{2}$ $0 ⩽ x ⩽ n a n d 0 ⩽ y ⩽ n \Rightarrow 0 ⩽ x^{2} + y^{2} ⩽ 2 n^{2} \Rightarrow 0 ⩽ r ⩽ n \sqrt{2}$ $θ \in [0, \frac{π}{2}] d i f f e m o r p h i s m e i s (r, θ) \to (x, y) = (φ_{1}, φ_{2}) (r c o s θ, \frac{r}{\sqrt{3}} s i n θ)$ $M_{j} (φ) = (\begin{matrix} \frac{\partial φ_{1}}{\partial r} \frac{\partial φ_{1}}{\partial θ} \\ \frac{\partial φ_{2}}{\partial r} \frac{\partial φ_{2}}{\partial θ} \end{matrix}) = (\begin{matrix} c o s θ - r s i n θ \\ \frac{s i n θ}{\sqrt{3}} \frac{r}{\sqrt{3}} c o s θ \end{matrix})$ $\Rightarrow d e t (M_{j}) = \frac{r}{\sqrt{3}} \Rightarrow A_{n} = \int_{0}^{\frac{π}{2}} \int_{0}^{n \sqrt{2}} e^{- r^{2}} s i n (r^{2}) \frac{r}{\sqrt{3}} d r d θ$ $= \frac{π}{2 \sqrt{3}} \int_{0}^{n \sqrt{2}} r e^{- r^{2}} s i n (r^{2}) d r b y p a r t s u^{^{'}} = {r e}^{- r^{2}} a n d v = s i n (r^{2})$ $A_{n} = \frac{π}{2 \sqrt{3}} {{[- \frac{1}{2} e^{- r^{2}} s i n (r^{2})]}_{0}^{n \sqrt{2}} - \int_{0}^{n \sqrt{2}} (- \frac{1}{2} e^{- r^{2}} \times 2 r c o s (r^{2})) d r}$ $= \frac{π}{2 \sqrt{3}} {- \frac{1}{2} e^{- 2 n^{2}} s i n (2 n^{2}) + \int_{0}^{n \sqrt{2}} r e^{- r^{2}} c o s (r^{2}) d r}$ $= \frac{π}{2 \sqrt{3}} {- \frac{1}{2} e^{- 2 n^{2}} s i n (2 n^{2}) + {[- \frac{1}{2} e^{- r^{2}} c o s (r^{2})]}_{0}^{n \sqrt{2}}$ $- \int_{0}^{n \sqrt{2}} (- \frac{1}{2} e^{- r^{2}} (- 2 r s i n (r^{2})) d r$ $= \frac{π}{2 \sqrt{3}} {- \frac{1}{2} e^{- 2 n^{2}} s i n (2 n^{2}) + \frac{1}{2} - \frac{1}{2} e^{- 2 n^{2}} c o s (2 n^{2}) - \int_{0}^{n \sqrt{2}} r e^{- r^{2}} s i n (r^{2}) d r}$ $\Rightarrow 2 A_{n} = \frac{π}{4 \sqrt{3}} {1 - e^{- 2 n^{2}} s i n (2 n^{2}) - e^{- 2 n^{2}} c o s (2 n^{2})} \Rightarrow$ $A_{n} = \frac{π}{8 \sqrt{3}} {1 - e^{- 2 n^{2}} s i n (2 n^{2}) - e^{- 2 n^{2}} c o s (2 n^{2})}$ $w e h a v e {l i m}_{n \to + \infty} e^{- 2 n^{2}} s i n (2 n^{2}) = {l i m}_{n \to + \infty} e^{- 2 n^{2}} c o s (2 n^{2}) = 0 \Rightarrow$ ${l i m}_{n \to + \infty} A_{n} = \frac{π}{8 \sqrt{3}}$
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