Problem 1 - Calculus Early Transcendentals

Question

Draw the region \(\\{(r, \theta): 1 \leq r \leq 2,0 \leq \theta \leq \pi / 2\\} .\) Why is it called a polar rectangle?

Step-by-Step Solution

Answer

Answer: A polar rectangle is a region in polar coordinates bound by two circles and two straight lines starting from the origin, which corresponds to the angular boundaries. It closely resembles a rectangle in Cartesian coordinates. To identify a polar rectangle from given inequalities, analyze the conditions on \(r\) and \(\theta\), sketch the region, and ensure constant radial and angular spans are present.

Step 1: Analyzing the inequalities

The inequalities provided are \(1 \leq r \leq 2\) and \(0 \leq \theta \leq \pi / 2\). The first inequality tells us that the region must lie between the circles with radius \(1\) and \(2\). The second inequality tells us that the region should be in the first quadrant due to the values of \(\theta\).

Step 2: Sketching the region

Sketch a \(r-\theta\) plane, and draw both the circles with radius \(1\) and \(2\). Also, mark the angle \(\theta = 0\) (positive x-axis) and \(\theta = \pi / 2\) (positive y-axis). The region described will be enclosed by these boundaries, which will form a sector in the first quadrant.

Step 3: Visualizing the polar rectangle

Now, due to the boundaries, it is clear that the region formed is a sector of a circle located in the first quadrant. The important characteristic of this region is that its radial and angular spans are constant. This means that the region is bound by two circles and two straight lines starting from the origin (corresponding to the angular boundaries). This shape closely resembles a rectangle in Cartesian coordinates - hence it is named a polar rectangle.