I have a problem where i don't understand how the solutions manual is doing the steps.

Problem: Find the area of the zone of a sphere formed by revolving the graph of

about the y axis.

I think you meant this to be
so,

then, sorry the square root suppose to go over the whole equation.

usually at this point it is

The solutions manual shows

Likewise, this should be
Why did they do this?

And the next step is confusing..

Likewise, this should be
Where did the 2 come from?

And in the next step where did the x come from?