Verified answer

I have answered your ather question.- Since x>0 y>0 and z>0 ( I octant),

0<φ<pi/2 ( A semisphere , I Octant )

0<θ<pi/2

0<r<2

Now , suppose z>0 only ,

0<φ<pi/2 ( A semisphere )

0<θ<2pi

0<r<2

Think what is dV .- dV = ( (rsinφ)dθ ) (This is one curvilinear side of dA) , then

the other curvilinear side ( rdφ) and finally depht dr , so dV = ( rsinφ)dθ ) ( ( rdφ)) dr

(rsinφ) is the projection of r over XY plane , ie, it is normal to Z axis , thus in a (1/4) of semisphere ( I Octant) you cover with (rsinφ) 0<θ<pi/2 , in a whole semisphere with (rsinφ) you cover 0<θ<2pi .- When you are doing this , in a(1/4) sphere or in a semisphere , ie , z>0 , you cover with the other side rφ in 0<φ<pi/2 .- If you have a whole sphere

0<φ<pi .- Depth is dr always .-

So a whole sphere

0<θ<2pi

0<φ<pi

0<r<R

A half ( z>0)

0<θ<2pi

0<φ<pi/2

0<r<R

A half sphere y>0 ( Y to the right , Z upward)

0<θ<pi

0<φ<pi

0<r<R

φ is measured from Z axis to XY plane , this is φ+, θ is measured counter clockwise , this is θ+

That is all , simply no more .- I hote this helps you .-

The notation changes depending on your source, so asking for the variable is not helpful. That's why good sources typically use azimuth and either elevation or zenith to distinguish. Azimuth is in the x-y plane and runs 0 to 2*pi. Elevation/zenith are the other typical angle (zenith is down from the z-axis, elevation is up from the x-y plane).

For your angles, the azimuth runs from 0 to pi/2 and zenith runs 0 to pi (elevation -pi/2 to pi/2). Phi vs. phi vs. theta is entirely dependent on your source - some use phi for azimuth, some use theta.