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Chapter 9: Problem 7

Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions. $$1 \leqslant r \leqslant 2$$

Short Answer

Expert verified
The region is an annular ring between circles with radii 1 and 2 centered at the origin.

Step by step solution

01

Understanding Polar Coordinates

Polar coordinates describe a location in the plane using two values: the radius \( r \) and the angle \( \theta \). The radius \( r \) is the distance from the origin to the point, and \( \theta \) is the angle from the positive x-axis to the line connecting the origin to the point. In this problem, \( r \) is given a range from 1 to 2.
02

Interpret the Given Condition

The condition \( 1 \leq r \leq 2 \) describes all points that have a radius between 1 and 2 units from the origin. This means we are talking about a ring or annular region, where the inner radius is 1 and the outer radius is 2.
03

Sketching the Inner Circle

Start by sketching a circle with radius \( r = 1 \). This circle is centered at the origin and represents the inner boundary of the region. All points on this circle are exactly 1 unit away from the origin.
04

Sketching the Outer Circle

Next, sketch another circle with radius \( r = 2 \). This will be the outer boundary of the region. Points on this circle are 2 units away from the origin.
05

Shading the Region

The region of interest is between the two circles, meaning it includes points that are more than 1 unit away from the origin and less than or equal to 2 units away. Shade the area between the two circles to represent this annular region.
06

Label the Diagram

Label the inner circle as \( r = 1 \) and the outer circle as \( r = 2 \). Also, indicate that the shaded region is the solution where \( 1 \leq r \leq 2 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Radius
In polar coordinates, the radius, denoted by \( r \), is an essential concept. It measures the straight-line distance from the origin, which is the center or base point of the coordinate system, to any given point in space. Think of the radius like stretching a piece of string from a central point to wherever the dot on a graph may be positioned.
It's crucial to understand how the radius functions because it's not just a length but an indicator of positioning in a polar grid.
  • A radius of \( r = 0 \) means the point is at the origin.
  • Positive values of \( r \) extend outward from the origin.
  • In our example, we're examining radii between 1 and 2 units.
This range helps define the specific region of our annular section.
Annular Region
An annular region resembles a ring or donut, characterized by its two circles: an inner circle and an outer circle. This is defined precisely by the condition \( 1 \leq r \leq 2 \) in polar coordinates. Conditions like these specify the part of the graph where only points having a radius between given values are included.
Imagine two concentric circles (same center), where:
  • The inner circle has a radius of 1, forming the smaller boundary.
  • The outer circle has a radius of 2, creating the larger boundary.
The space trapped between these two circles is our annular region. All points within this band are more than 1 unit from the origin, but not more than 2 units. This understanding is critical when visualizing any ring-like regions in polar plots.
Sketching Circles
Sketching circles in polar coordinates begins by marking the origin and using it as a central point. From there, understanding the radius enables you to draw circles with accurate sizes. In a coordinate system:
  • The inner circle, \( r = 1 \), needs to be drawn first. Every point on this circle maintains a constant distance of 1 from the origin.
  • The outer circle, \( r = 2 \), is larger. Points on it are consistently 2 units out from the origin.
The crucial step in sketching annular regions is identifying and shading the area in between these two circles. This stepped approach visually differentiates the inner and outer boundaries while highlighting the meaningful area where the solution lies. Ensure to label each circle and the shaded region to enhance clarity and understanding of the concepts involved.

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Most popular questions from this chapter

Find the area of the region that lies inside both curves. $$r^{2}=2 \sin 2 \theta, \quad r=1$$

How are the graphs of \(r=1+\sin (\theta-\pi / 6)\) and \(r=1+\sin (\theta-\pi / 3)\) related to the graph of \(r=1+\sin \theta\) ? In general, how is the graph of \(r=f(\theta-\alpha)\) related to the graph of \(r=f(\theta) ?\)

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