Why is the s-orbital always spherical in shape?
Because its wave function has no angular dependence.
By definition, an
Some explicit wave functions for the hydrogen atomic orbitals are:
#psi_(1s)(r,theta,phi) = 1/(sqrtpi) (1/a_0)^(3//2) e^(-r//a_0)#
#psi_(2s)(r,theta,phi) = 1/(4sqrt(2pi)) (1/a_0)^(3//2) (2 - r/a_0)e^(-r//2a_0)#
#psi_(3s)(r,theta,phi) = 1/(81sqrt(3pi)) (1/a_0)^(3//2) [27 - 18(r/a_0) + 2(r/a_0)^2]e^(-r//3a_0)#
The main thing you should notice is that all of these
That means there is no way the angles could deviate from a straight integration in spherical coordinates at a constant radius (giving a spherical integration path).