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

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Homework Help Overview

The discussion revolves around evaluating a double integral involving the expression \(\sqrt{x^2 + y^2}\) over the unit square region defined by \([0,1] \times [0,1]\) using a change of variables approach.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • The original poster attempts to use a change of variables by setting \(u = x^2\) and \(v = y^2\), calculating the Jacobian, and determining the limits of integration. Some participants suggest that converting to cylindrical coordinates might be easier, while others point out potential issues with the region of integration.

Discussion Status

The discussion is ongoing, with some participants questioning the correctness of the Jacobian calculation and the appropriateness of the chosen variable transformation. There are also suggestions for alternative approaches, but no consensus has been reached regarding the best method to evaluate the integral.

Contextual Notes

Participants note that the square region \([0,1] \times [0,1]\) does not correspond to the quarter of a circle, which raises questions about the validity of certain transformations. Additionally, there is mention of a previous post in another forum for further details.

ninjacookies
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I'm trying to evaluate the double integral

[tex]\int \int \sqrt{x^2 + y^2} \, dA[/tex]

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)

[tex]\int \int \sqrt{u + v} * (1) dudv[/tex]


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 [tex]x^2 + y^2[/tex] to [tex]r^2[/tex] 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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