I am working on a math problem that has me using Stokes' Theorem to calculate the surface area of hemisphere $x^2 + y^2 + z^2 = 1$, bounded by $y >= 0$. Graphing it out, it looks like a normal hemisphere cut in half at the Y axis.
When we learned how to infer the values of φ and θ for a parameterized sphere in class, we were told that generally φ is the vertical angle and θ is the horizontal angle. Of course, this varies per problem, but gives a decent idea as to how to approach it.
Using this, I inferred that since the y-axis splits the sphere in half, the value of φ is $π$, and since the hemisphere forms a shadow of a full circle on the x-z axis the value for θ is $2π$.
However, this is not the correct answer. The solution states that the integral goes from $0$ to $π$ for both φ and θ. Why is this? If the x-z axis forms a full circle, shouldn't it be $2π$?
Your hemisphere parametrization could be as
$$x=\sin (\phi)\cos (\theta) $$
$$y=\sin (\phi)\sin (\theta) $$
$$z=\cos (\phi) $$
with $$\theta \in [\frac {\pi}{2},\frac{3\pi}{2}] $$
and $$\phi\in [0,\pi]. $$