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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-




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
letAn=[0,n[2ex2y2x2+y2dxdy1)calculsteAnintermofn2)findlimn+An
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)
1)wedothechangement{x=rcosθy=rsinθ0x<nand0y<n0x2+y2<2n20r2<2n20r<n2An=0n20π2er2rrdrdθ=π20n2er2dr2)limn+An=π20er2dr=π2×π2=ππ4

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