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 ?

 

[nasu]

If you use x as the horizontall coordinate, the width of these strips is not dx.

 

[TensorCalculus]

It might also be more readable if you take some time to write it all out in TeX. Guide here. :)

 

[PeroK]

Maths and physics are hard enough without doing things the wrong way!

 

[kuruman]

Maths and physics are hard enough without doing things the wrong way!

Did you mean to say
Maths and physics are hard enough without trying to do things the harder way?