Applications of Double Integrals: Centroids and Symmetry

In summary, the problem involves finding the center of mass for a lamina occupying a specific region and with a density that is inversely proportional to its distance from the origin. The solution uses the symmetry of the region of integration and the density function to determine the center of mass. It is important to note that both the density function and the region must have symmetry in order for the solution to be valid.
  • #1
theBEAST
364
0

Homework Statement


A lamina occupies the region inside the circle x2+y2=2y but outside the circle x2+y2=1. Find the center of mass if the density at any point is inversely proportional to its distance from the origin.

Here is the solution:
https://dl.dropbox.com/u/64325990/MATH%20253/Centroids.PNG [Broken]

Why does it say by symmetry of the region of integration. Shouldn't it be by symmetry of the density function p(x,y) = k/root(x2+y2)?

For example what if our p(x,y) = x. Even though the region D is symmetric, the mass is no longer symmetric and the balancing point is no longer at x = 0. Am I right?

Thanks
 
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  • #2
theBEAST said:

Homework Statement


A lamina occupies the region inside the circle x2+y2=2y but outside the circle x2+y2=1. Find the center of mass if the density at any point is inversely proportional to its distance from the origin.

Here is the solution:
https://dl.dropbox.com/u/64325990/MATH%20253/Centroids.PNG [Broken]

Why does it say by symmetry of the region of integration. Shouldn't it be by symmetry of the density function p(x,y) = k/root(x2+y2)?

For example what if our p(x,y) = x. Even though the region D is symmetric, the mass is no longer symmetric and the balancing point is no longer at x = 0. Am I right?

Thanks
Yes, you are correct !
 
Last edited by a moderator:
  • #3
To reinstill faith in your textbook, it could be read as:

Symmetry of D,

and since f(x)=x.

Or even that f(x)=x is antisymmetric, perhaps a type of symmetry.
 
  • #4
You need the symmetry of both ##\rho(x,y)## and D. The density ##\rho(x,y)## is generally positive, so the only symmetry you can have is even symmetry. When multiplied by x, you get an odd integrand which then integrates to 0 because D is symmetric.
 

1. What is a double integral and how is it different from a single integral?

A double integral is a type of mathematical operation used to calculate the volume under a two-dimensional surface. It is different from a single integral, which is used to calculate the area under a one-dimensional curve. Double integrals involve integrating over a region in a two-dimensional plane, while single integrals are done over a one-dimensional interval.

2. How are double integrals used to find centroids and moments of inertia?

By using the properties of symmetry, double integrals can be used to find the centroid (center of mass) and moments of inertia (resistance to rotational motion) of a two-dimensional object. The centroid is found by taking the average of the x and y coordinates of the points in the region, while the moments of inertia are calculated by integrating over the region using the distance from the centroid as the radius.

3. Can double integrals be used to find the center of mass of irregularly shaped objects?

Yes, double integrals can be used to find the center of mass of any two-dimensional object, regardless of its shape. By dividing the object into small regions and calculating the centroid of each region, the overall centroid of the object can be found. This method can be applied to any shape, including irregular ones.

4. How can double integrals be used to find the area and volume of irregularly shaped objects?

Double integrals can be used to find the area and volume of any two-dimensional or three-dimensional object, respectively. By integrating over the region or solid, the total area or volume can be calculated. This is particularly useful for irregularly shaped objects or ones with curved surfaces.

5. Are there any real-world applications of double integrals?

Yes, there are many real-world applications of double integrals. They are commonly used in physics and engineering to calculate the volume or mass of objects, as well as to find the center of mass and moments of inertia. They are also used in economics to calculate the area under a demand or supply curve, and in statistics to find the probability of events occurring in a two-dimensional space.

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