Problem 8 - Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

Question: Sketch a polar rectangle with bounds 2 ≤ r ≤ 3 and π/4 ≤ θ ≤ 5π/4. Describe the steps you took in sketching this region. Answer: To sketch the polar rectangle R with bounds 2 ≤ r ≤ 3 and π/4 ≤ θ ≤ 5π/4, I first understood the constraints on the radius and angle values. I drew two lines at angles π/4 and 5π/4 to represent the θ boundaries. Then, I created two circles centered at the origin with radii of 2 and 3 for both angles to represent the r boundaries. Finally, I shaded the region enclosed by these lines and circles to represent the polar rectangle R.

Step 1: Understand the Constraints on r and θ

The polar rectangle R is defined by the bounds 2 ≤ r ≤ 3, and π/4 ≤ θ ≤ 5π/4. The bounds on r mean that the region R has a minimum radius of 2 and a maximum radius of 3. The bounds on θ mean that the region starts at an angle of π/4 radians (45 degrees) and goes to an angle of 5π/4 radians (225 degrees).

Step 2: Draw the θ-boundaries

In a polar coordinate system, we will first draw the angles π/4 and 5π/4 because these are the boundaries for θ. Draw a line representing angle π/4 from the origin to the edge of the graph, and another line representing angle 5π/4 from the origin to the edge of the graph. These two lines will form two sides of the polar rectangle.

Step 3: Draw the r-boundaries

Now, we need to draw the r constraints. At an angle of π/4, draw a circle with a radius of 2, centered at the origin. This will represent the minimum r value. Next, at the same angle, draw a circle with a radius of 3, also centered at the origin. This will represent the maximum r value. Repeat this process for the angle 5π/4. These circles will form the other two sides of the polar rectangle.

Step 4: Shade the Polar Rectangle

Finally, we can shade the region enclosed between the two θ lines and the two r circles. This shaded region represents the polar rectangle R, which satisfies the constraints 2 ≤ r ≤ 3, and π/4 ≤ θ ≤ 5π/4.