Application of double integrals: density

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SUMMARY

The discussion focuses on calculating the center of mass of a lamina defined by the semicircles \(y = \sqrt{1-x^2}\) and \(y = \sqrt{4-x^2}\), along with the x-axis segment connecting them. The density of the lamina is proportional to the distance from the origin, which is mathematically represented as \(\sqrt{x^2+y^2}\). The user initially struggled with determining the boundaries for integration but later realized that using polar coordinates simplifies the problem significantly.

PREREQUISITES
  • Understanding of double integrals in calculus
  • Knowledge of polar coordinates and their application in integration
  • Familiarity with the concept of center of mass in physics
  • Basic proficiency in mathematical functions and their graphical representations
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  • Study the application of double integrals in finding areas and volumes
  • Learn how to convert Cartesian coordinates to polar coordinates
  • Explore the concept of density functions and their implications in physics
  • Practice problems involving the center of mass for various shapes and densities
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Students studying calculus, particularly those focusing on double integrals and applications in physics, as well as educators looking for examples of density-related problems in laminae.

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Homework Statement



The boundary of a lamina consists of the semicircles y = sqrt(1-x^2) and y = sqrt(4-x^2) together with the portion of the x-axis that join them. Find the center of mass of the lamina if the density at any point is proportional to its distance from the origin

Homework Equations





The Attempt at a Solution



I have a hard time getting y and x boundaries, plus the function that's going to be integraded..
I understand that the function found be the distance from (0,0).. but how can I express that mathematically? I'm thinking that it could be simply sqrt(x^2+y^2), since x and y for a right trinagle, with hypotenuse being the distance to the point..

But with the boundaries - I'm completely lost.. help! :(
 
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Ok, never mind. I got it :) Forgot about the polar coordinates..
 

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