Application of double integrals: density

In summary, the problem involves finding the center of mass of a lamina with density proportional to its distance from the origin. The boundaries of the lamina consist of two semicircles and a portion of the x-axis. The function to be integrated is the distance from the origin, which can be expressed as sqrt(x^2+y^2) in polar coordinates.
  • #1
dlevanchuk
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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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  • #2
Ok, never mind. I got it :) Forgot about the polar coordinates..
 

FAQ: Application of double integrals: density

What is the purpose of using double integrals in density calculations?

Double integrals are used to calculate the total mass or density of a 3-dimensional object or region, by integrating over two variables. This allows for a more accurate representation of the distribution of mass or density throughout the object.

How do you set up a double integral for density calculations?

To set up a double integral for density calculations, you need to first define the limits of integration for both variables. This can be done by considering the boundaries of the object or region in question. Then, you need to determine the integrand, which is typically a function that represents the density at a given point. Finally, you can set up the double integral by integrating the density function over the defined limits.

Can double integrals be used to calculate the density of irregularly shaped objects?

Yes, double integrals can be used to calculate the density of irregularly shaped objects as long as the boundaries and density function can be defined mathematically. This allows for a more accurate calculation of the density compared to other methods such as using a single integral.

How does the order of integration affect the result of a double integral for density?

The order of integration can affect the result of a double integral for density, as it determines the direction in which the integration is performed. In some cases, one order may be easier to evaluate than the other, but the final result should be the same regardless of the order chosen.

Are there any real-world applications of double integrals in density calculations?

Yes, double integrals are commonly used in various fields such as physics, engineering, and economics to calculate the density or mass of objects or regions. For example, in fluid mechanics, double integrals are used to determine the volume and mass of a fluid flowing through a given region. In economics, double integrals can be used to calculate the average density of population in a city.

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