Answer
Please see the figure attached.
Work Step by Step
In spherical coordinates, $\rho$ is the distance back to the origin. Thus, $\rho=1$ describes a sphere of radius $1$. While $\phi$ is the angle of declination, which measures how much the ray through a point $P$ on the sphere declines from the vertical. Since it is in the interval $\frac{\pi }{3} \le \phi \le \frac{{2\pi }}{3}$, rotating it around the $z$-axis we obtain the part of the sphere as is shown in the figure.
