Menu Close

calculate-U-n-1-n-n-2-x-2-y-2-e-x-2-y-2-dxdy-and-lim-n-U-n-




Question Number 110451 by mathmax by abdo last updated on 29/Aug/20
calculate U_n =∫_([(1/n),n[^2 )     (x^2 −y^2 )e^(−x^2 −y^2 ) dxdy  and lim_(n→+∞)  U_n
calculateUn=[1n,n[2(x2y2)ex2y2dxdyandlimn+Un
Answered by mathmax by abdo last updated on 31/Aug/20
we consider the diffeomorphism   { ((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)  ⇒U_n =∫∫_(0≤r<n(√2) and 0≤θ≤(π/2))   r^2 (cos^2 θ−sin^2 θ)e^(−r^2 )  rdrdθ  =∫_0 ^(n(√2))    r^3  e^(−r^2 )  dr .∫_0 ^(π/2)   cos(2θ)dθ   but   ∫_0 ^(π/2)  cos(2θ)dθ =[(1/2)sin(2θ)]_0 ^(π/2)  =0 ⇒ U_n =0  ∀n ⇒lim U_n =0
weconsiderthediffeomorphism{x=rcosθy=rsinθ0x<nand0y<n0x2+y2<2n20r2<2n20r<n2Un=0r<n2and0θπ2r2(cos2θsin2θ)er2rdrdθ=0n2r3er2dr.0π2cos(2θ)dθbut0π2cos(2θ)dθ=[12sin(2θ)]0π2=0Un=0nlimUn=0

Leave a Reply

Your email address will not be published. Required fields are marked *