Problem 23 - Thomas Calculus

Question

Graph the sets of points whose polar coordinates satisfy the equations and inequalities in Exercises \(11-26 .\) $$\pi / 4 \leq \theta \leq 3 \pi / 4, \quad 0 \leq r \leq 1$$

Step-by-Step Solution

Answer

The region is a sector of a circle with radius 1, between angles \(\pi/4\) and \(3\pi/4\).

Step 1: Understand the Polar Coordinate System

Polar coordinates consist of an angle \(\theta\) and a radius \(r\). The angle \(\theta\) is measured from the positive x-axis, and \(r\) is the distance from the origin. The task is to graph the region defined by these constraints in the polar coordinate system.

Step 2: Identify Limits for \(\theta\)

Given \(\pi / 4 \leq \theta \leq 3 \pi / 4\), the angle \(\theta\) sweeps from \(\pi/4\) to \(3\pi/4\). This corresponds to the region between the lines \(\theta = \pi/4\) and \(\theta = 3\pi/4\) in the polar plane.

Step 3: Identify Limits for \(r\)

Given \(0 \leq r \leq 1\), the radius \(r\) is between 0 and 1. This means from the origin (\(r=0\)) to the circle with radius 1 (\(r=1\)). This ensures that we only consider points within a radius of 1 from the origin.

Step 4: Graph the Region

Combine these constraints: start drawing from \(\theta = \pi/4\) with \(r = 0\) to \(r = 1\), forming a straight line. Then, similarly, for \(\theta = 3\pi/4\). Finally, draw the arc of the circle from \(\theta = \pi/4\) to \(\theta = 3\pi/4\) at \(r = 1\). The shaded area between these boundaries forms a sector of a circle with two lines connecting the sector ends to the origin.

Key Concepts

Angle Measurement

In the polar coordinate system, angle measurement is a crucial concept. Instead of measuring angles in degrees like in the Cartesian system, we use radians. A full circle is equivalent to the angle measure of \(2\pi\) radians. This is important because the exercise specifies the angle \(\pi/4 \leq \theta \leq 3\pi/4\).

• \(\theta = \pi/4\): This represents an angle of 45 degrees.
• \(\theta = 3\pi/4\): This corresponds to an angle of 135 degrees.

These angles are measured from the positive x-axis, moving counter-clockwise in the polar plane. When sketching these, imagine rotating a line starting from the x-axis outward by the given angle. The region described in the exercise is therefore between these two angles understood in radians.

Radius in Polar Coordinates

The radius in polar coordinates, denoted by \(r\), represents the straight-line distance from the origin to a point in the polar plane. In the given exercise, the condition \(0 \leq r \leq 1\) establishes the range of possible radii.

• \(r = 0\): Represents the origin, the center of the polar coordinate system.
• \(r = 1\): Represents all points at a distance of 1 unit from the origin forming a circle.

The constraint given ensures that only the points within and along the boundary of this circle are considered. This means every point must be at most 1 unit away, starting from a simple point at the center. The consideration for \(r\) in this task creates a bounded region within the circle's edge itself.

Graphing in Polar Coordinates

Graphing in polar coordinates involves plotting points based on their angle \(\theta\) and radius \(r\). For this exercise, we aim to map out the region defined by \(\pi/4 \leq \theta \leq 3\pi/4\), and \(0 \leq r \leq 1\). Here's an overview of the graphing process:

• Begin at the line \(\theta = \pi/4\) with a radius \(r\) starting at 0 increasing up to 1. This line will stretch outward forming one boundary.
• Do the same with \(\theta = 3\pi/4\), creating another boundary.
• Connect these ends with an arc traced at \(r = 1\) between angles \(\pi/4\) and \(3\pi/4\).

The result is a sector of a circle. It's like a slice of pie, defined and enclosed by our angle and radius constraints. Visualizing this can make it easier to understand polar coordinates, bringing the abstract description to a more tangible concept.