[tex]0\le\theta\le\dfrac\pi2[/tex] tells you that [tex]x\ge0[/tex] and [tex]y\ge0[/tex].
Recalling that in cylindrical coordinates, [tex]r^2=x^2+y^2\implies r=\sqrt{x^2+y^2}[/tex], we then know that [tex]0\le\sqrt{x^2+y^2}\le z[/tex], and so we're confined to the first octant (where each of [tex]x,y,z[/tex] are non-negative).
The upper limit of [tex]z\le2[/tex] tells us that the region is bounded above by the plane [tex]z=2[/tex], which is parallel to the [tex]x[/tex]-[tex]y[/tex] plane.
Meanwhile, the lower limit of [tex]z=r=\sqrt{x^2+y^2}[/tex] can be visualized by first squaring both sides:
[tex]z=\sqrt{x^2+y^2}\implies z^2=x^2+y^2[/tex]
If you're not already familiar with what this equation represents, we can approach it piecemeal. At one extreme, when [tex]x=y=0[/tex], we have [tex]z=\sqrt{0^2+0^2}=0[/tex], so the region has a "vertex" at the origin.
When [tex]z=2[/tex], we have the Cartesian equation [tex]4=x^2+y^2[/tex], which corresponds to a circle of radius 2. Similarly, if we consider values of [tex]z[/tex] between 0 and 2, we end up with circles of increasing radii. Stacking these circles onto one another, we get a cone.
More specifically, the region is the part of the cone between the [tex]x[/tex]-[tex]y[/tex] plane and the plane [tex]z=2[/tex] restricted to the first octant.
(Image of region attached)
