Question Number 104773 by mathmax by abdo last updated on 23/Jul/20

let A_n =∫∫_([0,n[^2 )     (e^(−x^2 −y^2 ) /(√(x^2  +y^2 )))dxdy  1) calculste A_n  interm of n  2) find lim_(n→+∞) A_n

Answered by mathmax by abdo last updated on 24/Jul/20

1) we do the changement  { ((x =rcosθ)),((y =rsinθ)) :}  0≤x<n and 0≤y<n ⇒0≤x^2  +y^2 <2n^2  ⇒0≤r^2 <2n^2  ⇒0≤r<n(√2)  ⇒ A_n =∫_0 ^(n(√2))  ∫_0 ^(π/2) (e^(−r^2 ) /r) r dr dθ =(π/2) ∫_0 ^(n(√2))   e^(−r^2 ) dr  2) lim_(n→+∞)  A_n =(π/2) ∫_0 ^∞  e^(−r^2 ) dr =(π/2)×((√π)/2) =((π(√π))/4)