Finding the Center of Mass of a Lamina with Proportional Density

[mrcleanhands]

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 portions of the x-axis that join them. Find the centre of mass of the lamina if the density at any point is proportion to its distance from the origin.

Homework Equations

The Attempt at a Solution

My issue here is only in turning the statement "the density at any point is proportion to its distance from the origin" into a function.

The solution is $f(x,y)=k\sqrt{x^{2}+y^{2}}$ but why are they ignoring Z here? Since the lamina is in 3D right?

If I try turn this statement into a function I get $f(x,y) = k\sqrt{f(x,y)^{2} + x^{2} + y^{2}}$ which doesn't work. Where have I gone wrong in my thinking?

 

[mfb]

The shape is given as a 2D-object. How did you get the second equation, and what does it represent?