Help integrating Abs(x-y) dydx

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Discussion Overview

The discussion revolves around the integration of the function abs(x-y) over the unit square defined by the limits 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1. Participants explore methods to approach the problem, including the use of absolute value properties and region splitting.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Technical explanation

Main Points Raised

  • One participant suggests splitting the integral into two regions based on the relationship between x and y, specifically where x > y and x < y.
  • Another participant expresses a desire for a closed-form answer, indicating that the problem is to find a value B such that B∫[0,1]∫[0,1] abs(x-y) dydx = 1.
  • A participant questions the understanding of the absolute value function, implying that knowledge of its properties is essential for solving the integral.
  • It is noted that the region of integration can be visualized as a square, with the diagonal separating the areas where x ≥ y and y > x.
  • Participants emphasize the need to integrate the two regions separately and then combine the results.

Areas of Agreement / Disagreement

There is no consensus on the solution method, as participants present different approaches and levels of understanding regarding the integration process.

Contextual Notes

Some participants express uncertainty about the absolute value function, which may affect their ability to solve the integral. The discussion does not resolve the mathematical steps necessary for the integration.

mnf
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integrate:


∫01∫01 abs(x-y) dydx
 
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Welcome to PF!

Hi mnf! Welcome to PF! :smile:
mnf said:
integrate:


∫01∫01 abs(x-y) dydx


(no need to shout! :rolleyes:)

Hint: split the integral into two regions, one with x > y, and one with x < y. :wink:
 


I don't know how ,please explain it
 
Instead of one integral, with limits 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1 (which is a square),

split the square into two regions, one with x < y, and one with x > y,

and then use two integrals, one for each region.
 


i want to get answer in closed form
because question is
find B value

B∫01∫01 abs(x-y) dydx =1
 


If I didn't know better, I'd think you want us to solve it for you!
 


mnf said:
integrate:


∫01∫01 abs(x-y) dydx

If you honestly do not know what the absolute value of a number is, which is what you appear to be saying, you have no hope of doing this problem. Talk to your teacher about it!

If you do know the absolute value function then you know that |x- y|= x- y as long as x\ge y and |x-y|= y- x if x< y. The region 0\le x\le 1, 0\le y\le 1 is a square. x\ge y below the diagonal from (0, 0) to (1, 1) and y> x above the diagonal. Integrate those two separately and add.
 

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