Problem 35 - Precalculus with Calculus Previews

Question

Sketch the region on the plane that consists of points \((r, \theta)\) whose polar coordinates satisfy the given conditions. $$ 0 \leq r \leq 2,-\pi / 2 \leq \theta \leq \pi / 2 $$

Step-by-Step Solution

Answer

The region is a semicircle in the right-half plane bounded by radius 2.

Step 1: Understand Polar Coordinates

Polar coordinates are represented as \((r, \theta)\), where \(r\) is the distance from the origin, and \(\theta\) is the angle from the positive x-axis. The range \(0 \leq r \leq 2\) means the points are within a circle centered at the origin with a radius of 2. The range \(-\pi/2 \leq \theta \leq \pi/2\) means the angle \(\theta\) varies from \(-90^\circ\) to \(90^\circ\), covering the right half of the plane.

Step 2: Sketch the Circle with Center at Origin

Draw a circle with center at the origin and radius 2 on the polar plane. This circle includes all points \((r, \theta)\) where \(r\) is between 0 and 2. The boundary of the circle corresponds to the equation \(r = 2\).

Step 3: Identify the Angular Region

The angle \(\theta\) varies from \(-\pi/2\) to \(\pi/2\), which covers the right half of the polar coordinate plane. This means you consider angles starting from the negative y-axis, passing through the positive x-axis, and ending at the positive y-axis.

Step 4: Combine Conditions to Sketch the Region

Combine the circle with radius 2 and the angular range to sketch the region. This is the semicircle with radius 2 centered at the origin in the first and fourth quadrants of the Cartesian plane. It includes the boundary (circle) itself as well as all the interior points.

Key Concepts

Polar Plane

When working with polar coordinates, we often represent points on what is known as the polar plane. Imagine this plane similarly to the Cartesian plane, where we find points using an x and y coordinate. However, in the polar plane, each point is represented by two values:

• A distance from the origin, denoted by \( r \).
• An angle, denoted by \( \theta \), that is measured from the positive x-axis.

Instead of moving right, left, up, or down as we do in Cartesian coordinates, we move around circles at different radii and angles. This makes the polar plane useful for describing circular or rotational worlds. By varying these two values, \( r \) and \( \theta \), we can describe any point on this plane efficiently through rotations and spirals around the pole, which is the origin (point (0,0) in Cartesian, or \( r = 0 \) in polar coordinates).
To get started with understanding sketches on the polar plane, always think about how distance and angle are interacting to form the paths and shapes you expect to see.

Angular Region

An angular region in polar coordinates refers to the section of the plane that relates to specific angles \( \theta \). This is similar to how we define sector wedges like in a pie chart. Angle \( \theta \) defines the direction of a point from the positive x-axis.
For instance, consider an angular region defined by \(-\pi/2 \leq \theta \leq \pi/2\), which translates to angles between \(-90^\circ\) and \(90^\circ\). This specific region covers a half-circle from the bottom of the y-axis (negative y-axis) through the x-axis, to the top (positive y-axis).

• Points within this region can have any distance \( r \) from the origin, as long as they fall within these angular boundaries.
• This type of division is particularly useful for understanding symmetries and problems involving half rotations, like parts of circular problems limited to certain directions.

Identifying your target angular region helps in coordinating any circular shape you might need to sketch or understand – this is especially important when combining it with radius constraints.

Circle Sketching

Circle sketching in polar coordinates is a fundamental skill. Here, the idea is to be able to plot points that form circles based on their distance (\( r \)) from the origin. The equation \( r = 2 \) in polar coordinates describes a circle with a radius of 2 centered at the origin. This circle is fundamentally the set of all points that maintain a constant distance, or radius, from the origin.
To sketch such circles:

• Start by noting where the center of your circle is, which for polar coordinates is always the origin (\( r = 0 \)).
• Identify the radius – in this example, it’s 2, so every point on this circle will be exactly 2 units away from the origin at any given angle \( \theta \).

If you add angular restrictions, such as \(-\pi/2 \leq \theta \leq \pi/2\), you get only part of the circle, specifically a semicircle in this case. Understanding such constraints is key to crafting a precise sketch of the region of interest. A circle or circular segment in polar coordinates is not just about drawing a shape but incorporating both distance and directional constraints, resulting in an elegant representation of the problem's solution on the plane.