Numerically Integrating Function Over Sphere Surface Using Irregular Differentials

Numerically Integrating Function Over Sphere Surface Using Irregular
Differentials

I'm trying to (numerically) integrate a scalar-valued function over the
surface of a unit sphere with irregularly spaced differentials. I take
mathematical standard spherical coordinates:$$x=sin(\phi)cos(\theta)\\
y=sin(\phi)sin(\theta)\\ z=cos(\phi)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$
I need to integrate with $z$ and $\theta$ linear. This means that $\phi$
is nonlinear, of course.
I'm not really sure how I'd go about doing this. Using the Jacobian
determinant, I can get the spherical coordinate area element of $sin(\phi)
d\phi d\theta$, but using $sin(\phi) \Delta\phi \Delta\theta$ doesn't work
since it doesn't take into account $\phi$'s nonlinearity. Maybe I should
replace $\phi$ with $arccos(\phi)$ in the spherical coordinate definition
above and just try the Jacobian again?