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Chapter 16: Problem 10

Sketch the following polar rectangles. $$R=\\{(r, \theta): 4 \leq r \leq 5,-\pi / 3 \leq \theta \leq \pi / 2\\}$$

Short Answer

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Question: Sketch the polar rectangle given by the set of polar coordinates (饾憻,饾渻) such that 4 鈮� 饾憻 鈮� 5 and 鈭掟潨�/3 鈮� 饾渻 鈮� 饾湅/2, and describe the boundaries and enclosed region. Answer: The polar rectangle is bounded by two circles with radii 4 and 5, and two lines with angles -饾湅/3 and 饾湅/2. The enclosed region represents all the coordinates (r, 胃) such that 4 鈮� r 鈮� 5 and -饾湅/3 鈮� 饾渻 鈮� 饾湅/2.

Step by step solution

01

Identify the boundaries of the polar rectangle

We have a polar rectangle with the bounds \(4 \leq r \leq 5\) and \(-\pi / 3 \leq \theta \leq \pi / 2\). This means that the rectangular region is bound by two circles with radii 4 and 5, and two lines with angles \(-\pi / 3\) and \(\pi / 2\).
02

Sketch the circle with radius 4

Draw a circle centered at the origin (0,0) with radius 4. This circle represents the lower bound of the radial distance \(r\), which is all the points with \(r=4\).
03

Sketch the circle with radius 5

Draw another circle centered at the origin (0,0) with radius 5. This circle represents the upper bound of the radial distance \(r\), which is all the points with \(r=5\).
04

Sketch the line with angle \(-\pi/3\)

Next, draw a line starting at the origin that makes an angle of \(-\pi/3\) with the positive x-axis. This line represents the lower bound of the angle \(\theta\), which is all the points with \(\theta=-\pi/3\).
05

Sketch the line with angle \(\pi/2\)

Similarly, draw a line starting at the origin that makes an angle of \(\pi/2\) with the positive x-axis. This line represents the upper bound of the angle \(\theta\), which is all the points with \(\theta=\pi/2\).
06

Shade the polar rectangle region

Finally, shade the region that is enclosed by the two circles and the two lines to represent the polar rectangle. This shaded region represents all the coordinates \((r, \theta)\) such that \(4 \leq r \leq 5\) and \(-\pi / 3 \leq \theta \leq \pi / 2\).

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