Understanding what integrating in polar gives you

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    Integrating Polar
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Discussion Overview

The discussion revolves around the understanding of integration in polar coordinates, particularly focusing on the integral of the function \( R(x^2 - y^2) \) over a specified region in the first quadrant. Participants explore the visualization of the integral and the implications of symmetry in the function.

Discussion Character

  • Exploratory
  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • One participant expresses confusion about the result of the integral being zero despite the visual representation suggesting a non-zero area.
  • Another participant notes that the function \( x^2 - y^2 \) is symmetric in the region when reflected over the line \( x = y \), indicating that the function takes on both negative and positive values in that area.
  • A follow-up question seeks clarification on the significance of the line \( x = y \) in relation to the symmetry of the function.
  • Another participant hints at considering the nature of the squared terms in the function to understand the solutions better.

Areas of Agreement / Disagreement

Participants generally agree on the symmetry of the function and its implications for the integral, but the discussion remains unresolved regarding the participant's initial confusion about the visual representation versus the mathematical result.

Contextual Notes

The discussion does not resolve the underlying assumptions about the integration process in polar coordinates or the implications of symmetry on the integral's value.

tnmann10
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I am not understanding integration with polar coordinates, or at least visualizing what is happening. Here's the integral calculated in Wolfram:

http://www.wolframalpha.com/input/?i=integrate+(r^2(cost^2-sint^2))r+drdt+t=(0)..(pi/2)+r=(1)..(2)+

the integral before I changed it to polar was just ∫R(x2-y2)dA where R is the first quadrant region between the circles of radius 1 and 2

mathematically it makes sense that the answer is 0.

but when you draw the picture it is a quarter of a washer in the xy plane. This does not seem like 0 to me. Would someone please explain where my thinking is going wrong?

Thanks

First post:smile:
 
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The function x^2 -y^2 is symmetric in that region when you reflect over the line x=y, with the function negative on one half and positive on the other
 
ok that makes sense, but why the line x=y?
 
tnmann10 said:
ok that makes sense, but why the line x=y?

Think about the nature of ^2 term in terms of the solutions (hint: + and -).
 

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