Different way to find center of mass of Hollow hemisphere

[BlackUniverse]

Homework Statement

I want to find out center of mass of Hollow hemisphere using a vertical half ring element [Pls i know how to find it using a horizontal ring]

Relevant Equations

All eulqutions are in the pic.

Here is how I tried to solve it but I don't know how I got formula for disc.
Before this I tried to solve for half disc using a vertical element and it worked but not for this. Why?

 

[nasu]

You may get better/more answers if you formulate a more specific question rather than just this "check my work" format with three pages of hand written formulas with not much explanations of what they mean. Marking hand-written homework is a very tedious job.

 

[kuruman]

Specifically, how do you get from equation (1) to equation (2)? (see figure).

In equation (1) symbol $y$ stands for "the y-coordinate of infinitesimally small element $dm$."

It seems that for $dm$ you are using a semi-circle of width $dx$ and radius $x$. In equation (2) what did you use for its y-coordinate and why ?