Another quick please tell me if my logic seems correct (change of variables)

AI Thread Summary
The discussion revolves around evaluating the double integral of the function √(x² + y²) over the region R = [0,1] x [0,1] using a change of variables. The user initially set u = x² and v = y², calculating the Jacobian as 1, which led to the same integration limits for u and v. However, they later realized the Jacobian was incorrectly computed and expressed confusion about the most efficient setup for the integral. Other participants suggested converting to cylindrical coordinates, but noted that the square region does not correspond to a quarter circle, complicating the transformation. The conversation highlights the importance of correctly calculating the Jacobian and understanding the geometric implications of the chosen variables.
ninjacookies
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I'm trying to evaluate the double integral

\int \int \sqrt{x^2 + y^2} \, dA

over the region R = [0,1] x [0,1]
using change of variables.

Well, after fooling around, I've got an answer. I set u = x^2, v =y^2, and then calculated the jacobian of T which was 1. The image transformation limits of integration for u and v turned out to be the same [0,1] x [0,1]

So I did the following calculation (both integrals going from 0 to 1)

\int \int \sqrt{u + v} * (1) dudv


which resulted in a value of roughly 3.238.

Does my logic and answer seem sound here? Thanks in advance.
 
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Converting to rectangular coordinates would probably be easier
 
That's cylindrical coordinates, sorry
 
Answered in Calculus and Analysis.
 
I'm totally at a loss here guys. I realized my Jacobian was computed wrong. Can someone please give me a clue as to what would be the most efficient integral setup? I'm completely dumbfounded. :( Thanks


edit: more in-depth post in the calculus forum, thanks
 
Last edited:
Dexter...you need to drop the intensity down a notch. And to the ninja, just convert x^2 + y^2 to r^2 and integrate over the same area in cylindrical.
 
He can't do that,the square [0,1]*[0,1] is not equivalent to the quarter of the circle you're implying...

Daniel.
 

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