Simple Polar Inequalities Examples

Write inequalities for r and θ describing the shaded region.

If we take any point in the shaded region, θ must lie between 0 and .

The appropriate bounds for θ are

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

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

Write inequalities for r and θ describing the shaded region.

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, so 3 ≤ r ≤ 4. 

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

If θ is between π and 2π then the region in question is contained in the third and fourth quadrants.

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.

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. The blue part of the image below is the region that we want.