Regions in spherical coordinates

Spheres, hemispheres and similar regions are all described best in spherical coordinates. Here's a few examples.

1.

The spherical shell with an inner radius 1 and outer radius 2, centered on the origin can be described by $1 \le \rho \le 2, 0 \le \phi \le \pi, 0 \le \theta \le 2\pi$.

Note that the angle $\phi$ only ranges from 0 to $\pi$ in order to cover the entire sphere. To see this, make a sphere of radius a by moving a units along the z-axis, then sweeping this arc down toward the -z-axis by rotating the line segment through $\pi$ radians, then take the semi-circle and rotate it through $2\pi$ radians in the $\theta$ direction.

2.

The portion of the spherical cap shown below can be described by $1 \le \rho \le 2, 0 \le \phi \le \pi/6, 0 \le \theta \le \pi/4$.