1-calculate-A-n-0-n-2-dxdy-x-2-y-2-4-2-find-lim-n-A-n-

Question Number 65399 by mathmax by abdo last updated on 29/Jul/19

1) c a l c u l a t e A_{n} = \int \int_{[0, n [^{2}} \frac{d x d y}{\sqrt{x^{2} + y^{2} + 4}}

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

Commented by mathmax by abdo last updated on 31/Jul/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 e (r, θ) \to (x, y) /

x = r c o s θ a n d y = r s i n θ w e h a v e 0 ⩽ x < n a n d 0 ⩽ y < n \Rightarrow

0 ⩽ x^{2} + y^{2} < 2 n^{2} \Rightarrow 0 ⩽ r^{2} < 2 n^{2} \Rightarrow 0 ⩽ r < n \sqrt{2}

A_{n} = \int \int_{0 ⩽ r < n \sqrt{2} a n d 0 ⩽ θ ⩽ \frac{π}{2}} \frac{r d r d θ}{\sqrt{r^{2} + 4}}

= \int_{0}^{n \sqrt{2}} \frac{r d r}{\sqrt{r^{2} + 4}} \int_{0}^{\frac{π}{2}} d θ = \frac{π}{2} {[\sqrt{r^{2} + 4}]}_{0}^{n \sqrt{2}} = \frac{π}{2} {\sqrt{2 n^{2} + 4} - 2}

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