Applications of Double Integrals: Centroids and Symmetry

[theBEAST]

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

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

 

[SammyS]

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

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 !

 

[algebrat]

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.

 

[vela]

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.