How to draw the domain of a three variable function?

Question

I have this function

$$f(x,y,z)= \sqrt x+\sqrt y+\sqrt z+\ln(4-x²-y²-z²)$$

I know the domain is

$$Df={(x,y,z);x≥0,y≥0,z≥0,x²+y²+z²<4}$$

I need to draw the domain under the form of a sphere, as represented below:

Now my question is , the equation which constructed the general sphere here is x²+y²+z²=4.

But how do i know which parts of the sphere to highlight using the constraints, or else that all x y and z ≥0?

Meaning, which parts of the sphere may correspond to this restrictions? How do i know what to highlight to show the domain ?

EDIT: THIS IS NOT A DUPLICATE. Please do not mark it as such.

Answer 1

The points

• $x^2+y^2+z^2 < 4$ are inside the sphere (excluding the points on the surface)
• $x,y,z\ge 0$ are those in the first octant

therefore the domain are points in the first octant and inside the sphere.