Solving double integral without integrating

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Homework Help Overview

The discussion revolves around evaluating a double integral of the function 2x over a circular region defined by the equation x² + (y-1)² = 1, specifically focusing on how to conclude that the integral equals zero without performing the integration.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • Participants explore the concept of symmetry in the context of the circular region and how it affects the evaluation of the integral. There are discussions about the symmetry about the origin and the point (0,1), and how these symmetries lead to cancellation of contributions to the integral.

Discussion Status

The conversation is actively exploring the implications of symmetry on the integral's value. Participants are clarifying their understanding of the symmetry involved and how it relates to the conclusion that the integral evaluates to zero.

Contextual Notes

There is a focus on the geometric properties of the circular region and the function being integrated, with some participants correcting each other's understanding of the symmetry involved.

kasse
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From an example in my book:

Int Int (2x) dxdy over R = 0

(R is the circe x^2+(y-1)^2=1)

How can one make this conclusion without integrating?
 
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Look at the symmetry. A circle is "symmetric about the origin": If (x,y) is in the circle then so is (-x, -y). That means that for each possible x value, you have the corresponding -x and so the "effect" of the two points will cancel out.
 
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It's actually symmetric about (0,1), but you can draw a similar conclusion to HallsofIvy's post
 
Right, sorry about that! If (x. 1+y) is in the circle, so is (-x, 1- y)!
 

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