Evaluating double integral - jacobian help

  • Thread starter bon
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  • #1
bon
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Homework Statement



trying to evaluate the double integral from 0 to infinity and 0 to infinity of [(x^2 + y^2)/1 + (x^2-y^2)^2]e^-2xy dxdy

using the coordinate transformation u=x^2-y^2 and v=2xy



Homework Equations





The Attempt at a Solution



so i calculated the jacobian which looks nice 4(x^2+y^2)..can see some canceling there

just can't see what the new limits will be...

thanks for any help
 

Answers and Replies

  • #2
tiny-tim
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Hi bon! :smile:

(have an infinity: ∞ and try using the X2 tag just above the Reply box :wink:)

When you're trying to find new 2D or 3D limits, just draw the region, and then mark it with the "contour lines" of the new variables.

In this case, the region is the whole first quadrant …

now draw some typical curves for u = constant and for v = constant …

check that (u,v) is single-valued, and just read off the diagram what the lowest and highest "contour lines" are. :wink:
 
  • #3
bon
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Ok thanks so i see that v goes from 0 to infinity...i just can't see u at the moment..

thanks
 
  • #4
tiny-tim
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u = 0 is the straight diagonal line …

what are the other curves for u = constant in the first quadrant? :wink:
 
  • #5
bon
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ahh hyperbolae..so would it be -infinity to + infinity?

thanks
 
  • #6
bon
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is this right?
 
  • #7
tiny-tim
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ahh hyperbolae..so would it be -infinity to + infinity?

thanks
let's see …

each lower "hyperbola" is a quarter of a hyperbola, starting at the x-axis and finishing "at infinity", close to the diagonal …

so it goes from xy = 0 to xy = ∞ (and the same for the upper "hyperbolas").

So yes, x2 - y2 goes from -∞ to ∞, and for each value of x2 - y2, xy goes from 0 to ∞. :smile:
 
  • #8
bon
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great thanks
 

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