Triple integrals in spherical coordinates, volume of octant

Question

So, the question is : $S$ is the part of the sphere $ρ=a$ cut by the planes $\theta=0$ and $\theta=\frac{\pi}{6}$ in the first octant. Find the volume of $S$. I am taking the integration limits as $0≤\theta≤\frac{\pi}{6}$ and $0≤\phi≤\frac{\pi}{2}$ and $0\le \rho \le a$. Is this correct? And why do we take the $\phi$ limits from $0$ to $\pi$? Why not ${2}{\pi}$? I mean the point can be in the fourth and third octants too.

Answer 1

Those limit values are correct.

$\phi$ is the angle from the positive $z$-axis, when $\phi=0$, we are pointing at the positive $z$-axis. When $\phi=\frac{\pi}2$, we are pointing at the horizontal plane. When $\phi = \pi$, we are already pointing at the negative $z$-axis. Hence, we do not go beyond $\pi$.