Limits of integration spherical coordinates

Question

I have seen a lot of exercises where they solve a triple integral using spherical coordinates. But I'm confused about the limits that one should use. For example when they integrate over a sphere like $x^2 + y^2 + z^2 = 4$ I do understand why the limit are $0 \leq \rho \leq 2$ , $0\leq \theta \leq 2\pi$, but I can't get why this one ends in $\pi$ and not in $2\pi$, $0 \leq \phi \leq \pi$.
Thank You!

Answer 1

$0 \leq \phi \leq \pi$ Otherwhise you would be counting the volume twice!

Indeed, fix $\ \theta$ then, the equations $0 \leq \rho \leq 2$ , $0\leq \phi \leq \pi$ describe a half circle that has "vertical" diameter (see picture below)

By making this half-circle turn around $0\leq \theta \leq 2\pi$ we get the full sphere

Answer 2

Stand with your arm held directly above your head. Pretend your arm has radius $\rho=2$. Now swing it through $0\le\phi\le\pi$. You ought to have made a semicircle, and now your arm is resting against your leg. Next, keep swinging your arm through $\phi$, but also turn full circle on the balls of your feet, $0\le\theta\le2\pi$. Your arm ought to have swept out a sphere.

If your arm had initially gone from $0\le\phi\le2\pi$, you would have swept out two spheres in the end.