Evaluating double integral - jacobian help

[bon]

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

 

[tiny-tim]

Hi bon!

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

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.

 

[bon]

Ok thanks so i see that v goes from 0 to infinity...i just can't see u at the moment..

thanks

 

[tiny-tim]

u = 0 is the straight diagonal line …

what are the other curves for u = constant in the first quadrant?

 

[bon]

ahh hyperbolae..so would it be -infinity to + infinity?

thanks

 

[bon]

is this right?

 

[tiny-tim]

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 ∞.

 

[bon]

great thanks