Question

$$\text {Sketch the following sets of points.}$$ $$\\{(r, \theta): 2 \leq r \leq 8\\}$$

Short answer

Question: Sketch the set of points described by the polar coordinates (r,θ) with the constraint 2≤r≤8, without any specific angle restrictions for θ. Answer: An annular region between two circles with radii of 2 and 8 around the pole, without any specific angle restrictions for θ.

Step-by-step solution

Understand the problem

Since the constraint only describes a condition for the radial coordinate \(r\), there is no restriction on the angle coordinate \(\theta\). So \(r\) must be between 2 and 8 for every angle \(\theta\) from \(0\) to \(2\pi\). Essentially, this means that we are to sketch a closed region in the polar plane with \(r\) values ranging from 2 to 8 for all possible values of \(\theta\).

Sketch the inner boundary

Let's start by sketching the inner boundary, which is the curve \(r=2\). Since there are no restrictions on \(\theta\), we can simply draw a circle with radius 2 around the origin (the pole). This circle represents the points in the plane where \(r=2\).

Sketch the outer boundary

Next, let's sketch the outer boundary, which is the curve \(r=8\). Similar to the inner boundary, this will also be a circle around the pole with a radius of 8 units, representing the points in the plane where \(r=8\).

Sketch the region between boundaries

We have the two circles representing our boundaries now. The region we are looking to sketch is the area between these two circles. In other words, we want to shade the region between the inner circle (\(r=2\)) and the outer circle (\(r=8\)). This shaded annular region will represent the given set of points, where \(2 \leq r \leq 8\) for all values of \(\theta\). Now you have successfully sketched the set of points described by the given polar coordinates constraint. The final sketch should show a shaded annular region between the two circles (with radii of 2 and 8 units) with no specific angle restrictions for \(\theta\).

Key concepts

Radial Coordinate

In polar coordinates, the radial coordinate, denoted as \(r\), represents the distance from the origin (or pole) to a point in the plane. Imagine standing at the center of a circle and measuring how far away a point is from you.
This measurement is always done along the line that connects the origin to the point. If you think about our original exercise, the radial coordinate must be between two specific values: 2 and 8.
That means every point must be at least 2 units away from the origin but no more than 8 units, allowing for all radial lines from the origin within this range to be considered.

• Radial Coordinate \(r\) is always a non-negative value
• It defines a unique circle at a fixed distance
• The origin in polar coordinates is where \(r = 0\)

Understanding \(r\) helps us determine that only those points fitting within both the inner and outer circle are part of our set.

Angular Coordinate

The angular coordinate, \(\theta\), in polar coordinates describes the direction in which you measure the distance \(r\) from the pole. You consider the angle from a fixed direction, usually the positive x-axis, and measure it counterclockwise.
In our specific exercise, there are no restrictions on \(\theta\). This means angles can range from \(0\) to \(2\pi\), a full circle, allowing us to span the entire plane around the origin. This results in a circular sweep around the origin.

• Angular Coordinate \(\theta\) is measured in radians (\(0 \leq \theta < 2\pi\))
• It affects the rotation around the origin, not the size of the circle
• Each value of \(\theta\) corresponds to a unique direction

Thus, understanding \(\theta\) confirms that we are considering all possible angles when describing our region between the two circles.

Annular Region

The term "annular region" refers to the area between two concentric circles, like a ring-shaped slice of the plane. In our original exercise, the annular region is defined by the values of \(r\) and \(\theta\). Specifically, \(r\) is restricted to values between 2 and 8 for all \(\theta\) values from \(0\) to \(2\pi\).
This creates a ring starting at 2 units from the origin and ending at 8 units.

• An annular region is non-continuous with respect to the origin
• It excludes the inner circle while encompassing all points up to the outer circle
• The width of the annulus is the difference in the radii (8 - 2 = 6)

Visualizing this helps clarify that all points within this range form a band encircling the midpoint, showing the complete set of included points.