Write inequalities for r and θ describing the given region.
Answer
It looks like
. Within those values of θ, the value of r can be anything nonnegative, so the appropriate bound is
0 ≤ r.
It would be possible to describe this region in a way that allowed r to be negative, but there's no need to
go to the extra trouble.
Example 2
Write inequalities for r and θ describing the given region.
Answer
The value of r can be anything greater than 2, therefore
2 ≤ r.
Since the region wraps all the way around, bounds for θ are optional. We could say
0 ≤ θ ≤ 2π
or we could decline to give any bounds for θ.
Example 3
Write inequalities for r and θ describing the given region.
Answer
Counting the rays and the rings, we see that
and
2 ≤ r ≤ 3.
Example 4
Sketch the region described by the given inequalities.
Answer
Since no bounds are given for r, this region will extend outward infinitely. The conditions on θ include all of the third quadrant.
Example 5
Sketch the region described by the given inequalities.
r < 4
Answer
The condition r < 4 describes the inside of a circle of radius 4, not including the boundary of the circle. Since no bounds are given for θ
the region wraps all the way around the origin.
Example 6
Sketch the region described by the given inequalities.
and 2 ≤ r ≤ 5
Answer
The conditions on θ and r describe the crust of a pizza slice lying below the positive x-axis.
. Within those values of θ, the value of r can be anything nonnegative, so the appropriate bound is
0 ≤ r.
It would be possible to describe this region in a way that allowed r to be negative, but there's no need to
go to the extra trouble.
and
2 ≤ r ≤ 3. 
and 2 ≤ r ≤ 5