Problem 12 - Calculus Early Transcendentals

Question

\(7-12\) Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions. $$r \geqslant 1, \quad \pi \leqslant \theta \leqslant 2 \pi$$

Step-by-Step Solution

Answer

Sketch a region outside the circle of radius 1 between \(\pi\) and \(2\pi\) angles.

Step 1: Understanding Polar Coordinates

Polar coordinates consist of \(r\), the distance from the origin, and \(\theta\), the angle from the positive x-axis. The condition \(r \geq 1\) means all points are at least 1 unit away from the origin, forming a region outside or on the edge of a circle of radius 1.

Step 2: Analyzing Angular Range

The angle \(\theta\) is given as \(\pi \leq \theta \leq 2\pi\) which represents all the angles from \(\pi\) (which is 180 degrees) to \(2\pi\) (which is 360 degrees, completing a full circle). This describes angles in the third and fourth quadrants of the standard Cartesian plane.

Step 3: Combining Conditions to Define the Region

The points satisfying both \( r \geq 1\) and \(\pi \leq \theta \leq 2\pi\) will lie outside or on a circle of radius 1, in the half-plane that stretches from the negative x-axis to the positive x-axis. This forms a semi-infinite region in the third and fourth quadrants.

Step 4: Sketching the Region

Begin by drawing a circle of radius 1 centered at the origin. Since \(r \geq 1\), shade the region outside this circle. Then, only shade the half of this region occupied by angles from \(\pi\) to \(2\pi\), which corresponds to the portion of the plane on or below the x-axis (third and fourth quadrants).

Key Concepts

Angle Range

In polar coordinates, the angle \(\theta\) is an essential aspect that helps describe a point's location around a circle. The angle range is typically given in terms of radians rather than degrees. In this exercise, we are focusing on the angle range of \(\pi \leq \theta \leq 2\pi\).

• \(\theta = \pi\) corresponds to \(180\) degrees and points completely to the left on the negative x-axis.
• \(\theta = 2\pi\) is equivalent to \(360\) degrees, representing a full circle back to the starting point on the positive x-axis.

Therefore, this angle range covers the entire semi-circle below the x-axis, including both the third and fourth quadrants. This understanding of the angle range is crucial for identifying the region we are interested in.

Intersection of Regions

When dealing with polar coordinates, finding the intersection of regions involves combining different constraints on \(r\) and \(\theta\). In this case, we have two main conditions:

• \(r \geq 1\): Points must lie outside or exactly on a circle of radius 1.
• \(\pi \leq \theta \leq 2\pi\): Points must fall within specific angles.

The intersection of these conditions identifies a portion of the plane outside the circle and beneath the x-axis. We restrict our focus to the half-plane defined by these angles, meaning only parts of the plane that meet both criteria are included. This is important because only regions satisfying all conditions are valid for our interest.

Circle Sketching

Sketching plays a key role in understanding polar coordinates and visualizing regions. To begin sketching, we first draw a circle centered at the origin with a radius of 1. This circle represents all points at exactly 1 unit from the origin. Next:

• Consider the inequality \(r \geq 1\), which means we need to shade the area outside and including this circle.
• Then, limit the shaded area to the specific angle range between \(\pi\) and \(2\pi\), focusing on the lower half of the plane.

The region represents the ‘greater than or equal to’ scenario where all appropriate points lie in these shaded areas. The sketch helps visually confirm the theoretical understanding from the equation.

Polar Inequalities

Polar inequalities offer a way to specify regions in the polar coordinate system. The inequality \(r \geq 1\) indicates that all points are at least a distance of 1 from the origin, establishing a region outside or on the circle of radius 1. Working with this inequality involves understanding that

• The ">=" sign means points on the circle boundary are included.
• This forms an entire plane outside a certain threshold (here, outside the circle).

In polar coordinates, combining inequalities for \(r\) and \(\theta\) aids in isolating precise areas of interest. These inequalities determine boundaries and behaviors of shapes within the coordinate plane—essential for sketching and region analysis.