How to find the domain of a function using spherical coordinates?

Question

Hey guys so I was wondering how to find the domain of a function using spherical coordinates..

For example, I take two functions:

F(x,y,x)=√x+√y+√z+ln(4-x²-y²-z²) whose domain is

D=((x,y,z):x≥0,y≥0,z≥0,x²+y²+z²<4)

and

F(x,y,z)=√1-x²-y²-z² whose domain is

D=((x,y,z),x²+y²+z²≤1)

On the first equation , i see on the answer sheet that the sphere on (x,y,z) coordinates is only shaded about one quarter, whilst on the other equation all the sphere on the (x,y,z) coordinates is shaded?

I have dyscalculia and I do not understand on what base i should highlight the full sphere or not to depict the domain .

Any simple and clear help on how to know what part of the sphere to highlight to show the domain would be great!

Thanks!

Edit: I attached the drawings.

Answer 1

The first one is correct, it is the full ball. But in the second case, it is only the first octant, so an eigth, and not a quarter!