Sketch the solid described by the given inequalities.

It's a good idea to have some intuition on the shapes of common functions in cylindrical coordinates.

One thing I find the most helpful is equations relating $r$ and $z$. Since there is no $theta$ relation, the functions $z = z(r)$ all describe rotationally symmetric surfaces. The idea is if you can plot $z$ vs. $r$ in 2D, the 3D result is that curve revolved around the $z$ axis. If you've done anything similar to surfaces and volumes of revolution in single-variable calculus, this is the same idea. F

or example, functions of the form

$$ z = ar^2 + b $$

Looks like a parabola on the plane, so in 3D it is a paraboloid

Functions of the form $$ z = ar $$

are diagonal lines revolved around the $z$-axis, so they're shaped like a cone.

When these functions are combined with limits on $theta$ (like in your second example), it restricts the angle of revolution. So $0 le theta le pi/2$ on makes a quarter revolution in the first quadrant, and in effect looks like quarter cone.