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Chapter 16: Problem 8

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

Short Answer

Expert verified
Question: Sketch the polar rectangle defined by the intervals 2 鈮� r 鈮� 3 and \(\frac{\pi}{4} \leq 胃 \leq \frac{5 \pi}{4}\). Briefly describe the process. Answer: To sketch the polar rectangle, follow these steps: 1. Understand the given ranges of radial distances (2 to 3) and angular values (\(\frac{\pi}{4}\) to \(\frac{5\pi}{4}\)). 2. Plot radial limits by drawing two concentric circles with radii of 2 and 3. 3. Plot angular limits by drawing two lines originating from the origin (0,0) with angles equal to \(\frac{\pi}{4}\) and \(\frac{5\pi}{4}\). 4. Shade the area between the two concentric circles and the two lines to represent the polar rectangle. 5. Label the sketch with the given polar rectangle description R={(r,胃): 2 鈮� r 鈮� 3, \(\frac{\pi}{4} \leq 胃 \leq \frac{5 \pi}{4}\)}. The completed sketch shows the polar rectangle bounded by the specified intervals of radial distances and angular values.

Step by step solution

01

Understand the ranges

In this step, briefly examine the given ranges of r (2 to 3) and 胃 (\(\frac{\pi}{4}\) to \(\frac{5\pi}{4}\)). Radial distances are positive values between 2 and 3, and the angle 胃 varies between the first and third quadrants.
02

Plot radial limits

Draw two concentric circles with radii of 2 and 3. These circles represent the limits of the radial distances in the polar rectangle.
03

Plot angular limits

Draw two lines originating from the origin (0,0) with angles equal to \(\frac{\pi}{4}\) and \(\frac{5\pi}{4}\). These lines represent the limits of the angular coordinate in the polar rectangle.
04

Shade the polar rectangle

The area between the two concentric circles and the two lines is the polar rectangle. Shade this area to indicate the polar rectangle's boundaries.
05

Label the polar rectangle

Write the given polar rectangle description R={(r,胃): 2 鈮� r 鈮� 3, \(\frac{\pi}{4} \leq 胃 \leq \frac{5 \pi}{4}\)} next to the shaded area to show that the sketch corresponds to the given polar rectangle. Now, the sketch of the polar rectangle is complete.

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