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Question Number 65398 by mathmax by abdo last updated on 29/Jul/19

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

2) d e t e r m i n e {l i m}_{n \to + \infty} A_{n}

Commented by mathmax by abdo last updated on 31/Jul/19

1) 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 (r, θ) \to (x, y) = (r c o s θ, \frac{r}{\sqrt{3}} s i n θ)

φ (r, θ) = (φ_{1}, φ_{2}) / φ_{1} (r, θ) = r c o s θ a n d φ_{2} (r, 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})

a n d {d e t M}_{j} (φ) = \frac{r}{\sqrt{3}} w e h a v e 1 ⩽ x < n a n d 1 ⩽ y < n \Rightarrow

1 ⩽ x^{2} < n^{2} a n d 3 ⩽ 3 y^{2} < 3 n^{2} \Rightarrow 4 ⩽ x^{2} + 3 y^{2} < 4 n^{2} \Rightarrow 2 ⩽ r < 2 n

A_{n} = \int \int_{2 ⩽ r < 2 n a n d 0 ⩽ θ ⩽ \frac{π}{2}} s i n (r^{2}) e^{- r^{2}} \frac{r}{\sqrt{3}} d r d θ

= \frac{π}{2} \int_{2}^{2 n} r s i n (r^{2}) e^{- r^{2}} d r

b y p a r t s u^{^{'}} = r s i n (r^{2}) a n d v = e^{- r^{2}} \Rightarrow

\int_{2}^{2 n} r s i n (r^{2}) e^{- r^{2}} d r = {[- \frac{1}{2} c o s (r^{2}) e^{- r^{2}}]}_{2}^{2 n} - \int_{2}^{2 n} (- \frac{1}{2} c o s (r^{2})) (- 2 r) e^{- r^{2}} d r

= - \frac{1}{2} {c o s (4 n^{2}) e^{- 4 n^{2}} - c o s (4) e^{- 4}} - \int_{2}^{2 n} r c o s (r^{2}) e^{- r^{2}} d r

a g a i n b y p a r t s u^{^{'}} = r c o s (r^{2}) a n d v = e^{- r^{2}} \Rightarrow

\int_{2}^{2 n} r c o s (r^{2}) e^{- r^{2}} d r = {[\frac{1}{2} s i n (r^{2}) e^{- r^{2}}]}_{2}^{2 n} - \int_{2}^{2 n} \frac{1}{2} s i n (r^{2}) (- 2 r) e^{- r^{2}} d r

= \frac{1}{2} {s i n (4 n^{2}) e^{- 4 n^{2}} - s i n 4 e^{- 4}} + \int_{2}^{2 n} r s i n (r^{2}) e^{- r^{2}} d r \Rightarrow

2 \int_{2}^{2 n} r s i n (r^{2}) e^{- r^{2}} d r = \frac{1}{2} {c o s (4) e^{- 4} - c o s (4 n^{2}) e^{- 4 n^{2}}}

+ \frac{1}{2} {s i n (4) e^{- 4} - s i n (4 n^{2}) e^{- 4 n^{2}}} \Rightarrow

A_{n} = \frac{π}{8} {e^{- 4} (c o s (4) + s i n 4) - e^{- 4 n^{2}} (c o s (4 n^{2}) + s i n (4 n^{2})}

2) {l i m}_{n \to + \infty} A_{n} = \frac{π e^{- 4}}{8} {c o s (4) + s i n (4)}