Write inequalities for r and θ describing the shaded region.

PICTURE polar ineqs 1

If we take any point in the shaded region, θ must lie between 0 and $\frac{π}{2}$.

PICTURE polar ineqs 2

The appropriate bounds for θ are
0≤θ≤ \frac{π}{2}.

Any point in the shaded region must have r between 0 and 1.

PICTURE polar ineqs 3

The appropriate bounds for r are
0≤ r ≤ 1.

Write inequalities for r and θ describing the shaded region. 

PICTURE polar ineqs 4

Since the region wraps all the way around the origin, we have 0≤ θ≤ 2π Actually, any value of θ is fine, but it's tidier to give bounds. The value of r goes from 3 to 4 inside the region, therefore 3≤ r≤ 4. 

PICTURE polar ineqs 5

Sketch the region described by the inequalities π ≤ θ ≤ 2 π 1 ≤ r ≤ 2.

If θ is between π and 2π then the region in question is contained in the third and fourth quadrants. PICTURE polar ineqs 10 Since r must be between 1 and 2, we shade in the portion of these quadrants which is at least 1 but not more than 2 from the origin: PICTURE polar ineqs 11

Sketch the region described by the inequality r>2.

Since no bounds are given for θ, the region will wrap all the way around the origin. The condition r > 2 describes the region outside a circle of radius 2, not including the boundary of that circle. PICTURE graph. Dashed line at circle r = 2, shade outside.