Finding mass using multiple integrals

In summary, the mass of a circular lamina of radius a whose density per unit area varies as the cube of the distance from a single point on the edge can be calculated by using the equation ρdxdydz and setting up suitable slices, such as arcs of thickness dr, before integrating.
  • #1
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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  • #2
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:
 

What is the definition of mass in terms of multiple integrals?

The mass of an object is the total amount of matter it contains. In terms of multiple integrals, mass can be defined as the triple integral of the density function over the volume of the object.

How is the density function used to find mass using multiple integrals?

The density function represents the amount of mass per unit volume at a given point in space. By integrating this function over the volume of the object, the total mass can be calculated.

What is the difference between single and multiple integrals when finding mass?

A single integral can be used to find the mass of a one-dimensional object, such as a wire. However, for three-dimensional objects, multiple integrals are needed to account for the mass in each dimension.

Can multiple integrals be used to find the mass of irregularly shaped objects?

Yes, multiple integrals can be used to find the mass of any object, regardless of its shape. This is because the volume of the object can be divided into infinitesimally small elements, making it possible to integrate over any irregular shape.

How does the choice of coordinate system affect the calculation of mass using multiple integrals?

The choice of coordinate system, such as Cartesian, cylindrical, or spherical, can affect the complexity of the integrals used to find mass. It is important to choose the most appropriate system for the given object to simplify the calculation process.

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