Problem 9 - Calculus Early Transcendentals

Question

Sketch the following polar rectangles. $$R=\\{(r, \theta): 1 \leq r \leq 4,-\pi / 4 \leq \theta \leq 2 \pi / 3\\}$$

Step-by-Step Solution

Answer

Answer: The polar rectangle is defined by the following four Cartesian coordinates: 1. \(x_1 = 1\cos(-\pi/4)\), \(y_1 = 1\sin(-\pi/4)\) 2. \(x_2 = 4\cos(-\pi/4)\), \(y_2 = 4\sin(-\pi/4)\) 3. \(x_3 = 4\cos(2\pi/3)\), \(y_3 = 4\sin(2\pi/3)\) 4. \(x_4 = 1\cos(2\pi/3)\), \(y_4 = 1\sin(2\pi/3)\)

Step 1: Convert Polar Coordinate Limits to Cartesian Coordinate Limits

Given the polar coordinate limits for r and θ, we must first convert them to their corresponding Cartesian coordinate limits (x and y). The conversions between polar and Cartesian coordinates are given by the following formulas: $$x = r \cos \theta$$ $$y = r \sin \theta$$ In this case, r ranges from 1 to 4, and θ ranges from \(-\pi / 4\) to \(2 \pi / 3\).

Step 2: Compute the Corners of the Rectangle in Cartesian Coordinates

Next, find the corners of the rectangle by using the Cartesian coordinate limits obtained in step 1. The four corners are given by: 1. The minimum x value and the minimum y value: \(x_1 = 1\cos(-\pi/4)\) and \(y_1 = 1\sin(-\pi/4)\) 2. The maximum x value and the minimum y value: \(x_2 = 4\cos(-\pi/4)\) and \(y_2 = 4\sin(-\pi/4)\) 3. The maximum x value and the maximum y value: \(x_3 = 4\cos(2\pi/3)\) and \(y_3 = 4\sin(2\pi/3)\) 4. The minimum x value and the maximum y value: \(x_4 = 1\cos(2\pi/3)\) and \(y_4 = 1\sin(2\pi/3)\)

Step 3: Plot the Rectangle on the Cartesian Plane

Now that we have found the Cartesian coordinates of the corners of the rectangle, we plot them on the Cartesian plane. Connect the points in the order given above and complete the rectangle.

Step 4: Add the Polar Coordinate Grid Lines

Lastly, add the polar coordinate grid lines to the plot we made in step 3. This can be done by drawing lines of constant r and lines of constant θ within the boundaries of the rectangle. Since r ranges from 1 to 4 and θ ranges from \(-\pi / 4\) to \(2 \pi / 3\), we will draw: 1. Lines of constant r: Circle arcs with radii 1, 2, 3, and 4 centered at the origin. 2. Lines of constant θ: The lines going through the origin with angles of \(-\pi / 4\) and \(2 \pi / 3\) radians. Now, you should have completed the sketch of the given polar rectangle, including the Cartesian representation and the polar coordinate grid lines.