Problem 26 - Thomas Calculus

Question

Graph the sets of points whose polar coordinates satisfy the equations and inequalities in Exercises \(11-26 .\) $$0 \leq \theta \leq \pi / 2, \quad 1 \leq|r| \leq 2$$

Step-by-Step Solution

Answer

The graph is a quarter-annulus in the first quadrant, between circles of radii 1 and 2.

Step 1: Understanding Polar Coordinates

Polar coordinates consist of a pair \(r, \theta\), where \(r\) is the radial distance from the origin and \(\theta\) is the angle from the positive x-axis. In this exercise, \(0 \leq \theta \leq \pi/2\) covers the first quadrant.

Step 2: Radial Distance Constraints

The inequality \(1 \leq |r| \leq 2\) means the radial distance must be between 1 and 2. This will form a ring-like region between circles of radii 1 and 2 centered at the origin.

Step 3: Graphing in the First Quadrant

Since \(0 \leq \theta \leq \pi/2\), we are only concerned with the first quadrant of the polar coordinate system. We will graph within \(r = 1\) and \(r = 2\), ensuring the region forms a quarter-ring or slice of an annulus.

Step 4: Combine Radial Distance and Angle

In combining both constraints \(1 \leq |r| \leq 2\) and \(0 \leq \theta \leq \pi/2\), draw the region with all points \(P(r, \theta)\) where \(r\) is between 1 and 2, and \(\theta\) is within the first quadrant.

Key Concepts

Radial Distance

In polar coordinates, the concept of radial distance is key, serving as one of the two coordinates that define a point's position. It is denoted by \(r\) and represents how far a point is from the origin, which is considered the center of the coordinate system. Unlike Cartesian coordinates, where we use two perpendicular lines to find a point’s position, polar coordinates use this distance to simplify understanding of circular and rotational systems.

• The radial distance \(|r|\) provides flexibility allowing for both inward and outward direction indicated by positive and negative values.
• In this particular exercise, the constraint \(1 \leq |r| \leq 2\) means that the points are found between two circles of radii 1 and 2.
• This creates an annular (ring-shaped) area, illustrating a unique feature of polar coordinates where circular dimensions are straightforward to represent.

Therefore, radial distance permits easy depiction and understanding of circular plots in polar graphs.

Angle Measure

Angle measure, symbolized by \(\theta\), is the second coordinate in polar coordinates and is equally vital. It specifies the rotation angle from the positive x-axis towards the point's direction around the origin. This is especially important in representing symmetrical shapes and directional graphs.

• The angle \(\theta\) is usually measured in radians, a natural unit of angular measure in mathematics.
• In the exercise, \(\theta\) ranges from \(0\) to \(\pi/2\), limiting our interest to quarter-circle portions.
• This limit helps direct the plot to display only the first section of a circular graph aligned with the positive x and y axes.

Thus, understanding angle measurement in polar coordinates aids in constructing and analyzing asymmetric and rotationally symmetrical shapes effectively.

First Quadrant

Focusing within the first quadrant of the polar coordinate system simplifies visualizing and conceptualizing simpler regions. Polar coordinates traditionally divide the plane into four sections based on angle measures. The first quadrant corresponds to \(\theta\) ranging from \(0\) to \(\pi/2\).

• Positioning here means the plotting covers angles that lie in the northeast direction from the origin.
• In this exercise, it involves graphing within the bounds of a small and large circle segment only one quarter deep.
• It provides a quarter-annulus shape, due to the combined radial and angular constraints.

Thus, working within the first quadrant allows for focused study on specific sections of circular graphs, making it easier to apply radial and angular limitations effectively.