let-A-n-w-n-e-x-2-y-2-x-2-y-2-dxdy-with-w-n-1-n-n-1-n-n-1-calculate-A-n-interms-of-n-2-find-lim-n-A-n-

Question Number 57103 by turbo msup by abdo last updated on 30/Mar/19

l e t A_{n} = \int \int_{w_{n}} e^{- x^{2} - y^{2}} \sqrt{x^{2} + y^{2}} d x d y

w i t h w_{n} = [\frac{1}{n}, n] \times [\frac{1}{n}, n]

1) c a l c u l a t e A_{n} i n t e r m s o f n

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

Commented by 121194 last updated on 30/Mar/19

x = r \cos θ

y = r \sin θ

J (r, θ) = r

\int \int_{w_{n}^{*}} e^{- r^{2}} r \cdot r d r d θ = \int \int_{w_{n}^{*}} e^{- r^{2}} r^{2} d r d θ

Commented by maxmathsup by imad last updated on 30/Mar/19

l e t u s e t h e c h a n g e m e n t x = r c o s θ a n d y = r s i n θ 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{2}{n^{2}} ⩽ r^{2} ⩽ 2 n^{2} \Rightarrow \frac{\sqrt{2}}{n} ⩽ r ⩽ n \sqrt{2}

a l s o w e h a v e x ⩾ 0 a n d y ⩾ 0 \Rightarrow 0 ⩽ θ ⩽ \frac{π}{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}} r r d r d θ = \int \int_{w^{^{'}} n} r^{2} e^{- r^{2}} d r d θ

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

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

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

A_{n} = \frac{π \sqrt{2}}{4} {\frac{1}{n} e^{- \frac{2}{n^{2}}} - n e^{- 2 n^{2}}} + \frac{π}{4} \int_{\frac{\sqrt{2}}{n}}^{n \sqrt{2}} e^{- r^{2}} d r \dots

2) w e h a v e {l i m}_{n \to + \infty} \frac{1}{n} e^{- \frac{2}{n^{2}}} - n e^{- 2 n^{2}} = 0 a n d {l i m}_{n \to + \infty} \int_{\frac{\sqrt{2}}{n}}^{n \sqrt{2}} e^{- r^{2}} d r

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