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Chapter 10: Problem 9

\(7-12\) Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions. $$0 \leqslant r<4, \quad-\pi / 2 \leqslant \theta<\pi / 6$$

Short Answer

Expert verified
The region is a wedge-shaped sector in the circle with radius less than 4, between \( -\pi/2 \) and \( \pi/6 \).

Step by step solution

01

Understanding Polar Coordinates

Polar coordinates describe a point in the plane using the distance from the origin, denoted by \( r \), and the angle \( \theta \) from the positive x-axis. In this problem, we need to identify the region defined by constraints on \( r \) and \( \theta \).
02

Interpreting the Condition on r

The condition \( 0 \leq r < 4 \) means we are looking at a region within a circle of radius 4, centered at the origin. This is a sector of a circle rather than the entire circle, as \( r \) cannot go beyond 4.
03

Interpreting the Condition on 胃

The condition \( -\frac{\pi}{2} \leq \theta < \frac{\pi}{6} \) implies an angular restriction. This corresponds to angles from the negative y-axis \( -\pi / 2 \) right up to \( \pi / 6 \) radians. It's an angle that opens counter-clockwise in standard mathematics conventions.
04

Visualizing the Region

Combine the conditions: The region is part of a circle with radius growing from the origin to less than 4, within the angle interval from \( -\frac{\pi}{2} \) (negative y-axis) to \( \frac{\pi}{6} \) (slightly positive of x-axis). This forms a wedge-like shape not including the circular arc at \( r=4 \).
05

Sketching the Region

To sketch the region, start by drawing a circle of radius 4, but keep in mind we're interested in the part inside it. Then, draw two lines: one, the negative y-axis as \( \theta=-\pi/2 \), and another at approximately 30 degrees to the positive x-axis as \( \theta=\pi/6 \). Shade the area enclosed by these lines and the circle centered at the origin with radius less than 4.

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Key Concepts

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

Radius Constraints
In polar coordinates, the radius plays a significant role in determining the position of a point. It measures the distance from the origin to the point in question. For our exercise, the radius is constrained by the inequality \(0 \leq r < 4\). This implies a few essential things:
  • The radius, \(r\), cannot be negative, meaning all points are at non-negative distances from the origin.
  • The maximum value of the radius is just below 4, indicating the region lies within a circle of radius 4 centered at the origin.
Understanding this constraint forms the foundation of sketching polar regions since it tells us how far our region extends from the origin. Curiously, even though it approaches a distance of 4, it never includes the circular boundary where \(r=4\). Instead, it resembles a circle without its outer edge.
Angular Constraints
The parameter \(\theta\) in polar coordinates describes the angle from the positive x-axis, dictating the direction of the radius. For our scenario, the angular constraint \(-\frac{\pi}{2} \leq \theta < \frac{\pi}{6}\) delineates the section of the plane we're interested in.
  • The angle of \(-\frac{\pi}{2}\) corresponds to pointing downward along the negative y-axis.
  • On the other hand, \(\frac{\pi}{6}\) translates roughly to 30 degrees in a counter-clockwise direction from the positive x-axis.
Combining these constraints gives us a wedge-shaped slice of the circle from angle \(-\frac{\pi}{2}\) to \(\frac{\pi}{6}\). This part excludes the line at \(\theta = \frac{\pi}{6}\) itself, but forms one boundary of our region.
Sketching Regions
Visualizing and sketching polar regions involves integrating both radius and angular constraints into one coherent representation.
  • Begin by sketching a circle with a radius of 4, which serves as the boundary for \(r\). Remember, the points within this boundary must lie inside the circle, excluding the line representing \(r=4\).
  • Next, illustrate the angular constraints by marking lines corresponding to angles \(-\frac{\pi}{2}\) and \(\frac{\pi}{6}\). These lines define the edges of the wedge.
To complete the sketch, shade the region enclosed by the lines and the circle under the radius constraint. Keep the shaded area approaching, but never reaching, the circle at radius 4. This depiction captures the limited range in terms of both distance and angle, resulting in a specific part of the plane adhering to both constraints.

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Most popular questions from this chapter

(a) Find the eccentricity, (b) identify the conic, (c) give an equation of the directrix, and (d) sketch the conic. $$r=\frac{3}{2+2 \cos \theta}$$

Write a polar equation of a conic with the focus at the origin and the given data. Ellipse, eccentricity \(\frac{3}{4}, \quad\) directrix \(x=-5\)

\(63-68\) Find the points on the given curve where the tangent line is horizontal or vertical. $$r=2+\sin \theta$$

\(71-76\) Use a graphing device to graph the polar curve. Choose the parameter interval to make sure that you produce the entire curve. $$r=2-5 \sin (\theta / 6)$$

(a) Show that the curvature at each point of a straight line is \(\kappa=0 .\) (b) Show that the curvature at each point of a circle of radius \(r\) is \(\kappa=1 / r=1 / r\).
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