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Changing the variable in multiple integrals 
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#1
Jan2209, 11:29 AM

P: 88

1. The problem statement, all variables and given/known data
Evaluate [tex]\int\int(xy)^2sin^2(x+y)dxdy[/tex] taken over a square with successive vertices (pi,0), (2pi,pi), (pi,2pi), (0,pi). 2. Relevant equations [tex]I = \int\int_{K} f(x,y)dxdy = \int\int_{K'} g(u,v)*J*dudv[/tex] where J is the Jacobian. 3. The attempt at a solution Okay so I've just been learning this for the first time, so I may be doing it completely wrong! I used the transformations u=xy, v=x+y which give the Jacobian as 2. Now i wasn't sure how to get the new limits for the integrals. What I did was apply the transformation above to give new vertices: (pi,0) > (pi,pi) (0,pi) > (pi,pi) (pi,2pi) > (pi,3pi) (2pi,pi) > (pi,3pi) This gives a simple rectangle, so then i just wrote [tex]I = 2*\int^{3\pi}_{\pi}\int^{\pi}_{\pi}u^2sin^2(v)dudv = \frac{4\pi^{4}}{3}.[/tex] I wish this was right, but i've a feeling it's not :( Any help greatly appreciated! 


#2
Jan2209, 01:47 PM

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The rectangle looks ok. But haven't you got the jacobian factor upside down?



#3
Jan2209, 02:31 PM

P: 88

Ah yeah, should be 1/2. Other than that though does my method look correct?
Thanks. 


#4
Jan2209, 02:34 PM

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P: 25,243

Changing the variable in multiple integrals



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