Finding mass using multiple integrals

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SUMMARY

The discussion focuses on calculating the mass of a circular lamina with a radius 'a', where the density varies as the cube of the distance from a point on the edge. The initial confusion regarding the absence of a density equation is clarified by recognizing that the density function is implicitly defined by the distance from the edge. The recommended approach involves using polar coordinates and integrating suitable slices of thickness 'dr' to find the mass.

PREREQUISITES
  • Understanding of multiple integrals
  • Familiarity with polar coordinates
  • Knowledge of density functions in physics
  • Basic calculus concepts, including integration techniques
NEXT STEPS
  • Study the application of polar coordinates in multiple integrals
  • Learn about density functions and their implications in mass calculations
  • Practice problems involving integration of variable density over geometric shapes
  • Explore the concept of slicing in integration for complex shapes
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Students studying calculus, particularly those focusing on multiple integrals and applications in physics, as well as educators looking for examples of density variations in mass calculations.

nb89
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The density per unit area of a circular lamina of radius a varies as the cube of the distance from a single point on the edge. Find the mass of this lamina.


im guessing id have to do ρdxdydz, and maybe use polar coordinates but I am completely lost. I am used to the question giving me an equation for the density which this doesn't have.

Any help would be much appreciated
(This is an exam question i had today but failed miserably at!)
 
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nb89 said:
The density per unit area of a circular lamina of radius a varies as the cube of the distance from a single point on the edge. Find the mass of this lamina.

im guessing id have to do ρdxdydz, and maybe use polar coordinates but I am completely lost. I am used to the question giving me an equation for the density which this doesn't have.

Hi nb89! :smile:

i] you do have an equation for the density … it's the cube of that distance

ii] in problems like this, choose suitable slices before integrating …

in this case, arcs of thickness dr :smile:
 

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