Problem 10 - Calculus Early Transcendentals

Question

\(7-12\) Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions. $$ 2

Step-by-Step Solution

Answer

A ring-shaped sector between circles of radius 2 and 5, spanning angles \(3\pi/4\) to \(5\pi/4\).

Step 1: Understanding Polar Coordinates

Polar coordinates are represented as \((r, \theta)\), where \(r\) is the distance from the origin, and \(\theta\) is the angle from the positive x-axis. The given conditions suggest that \(r\) ranges from greater than 2 to less than or equal to 5, and \(\theta\) is between \(3\pi/4\) and \(5\pi/4\).

Step 2: Describing the Region

The region described by \(2 < r \leq 5\) forms a circular annular sector. The outer radius (boundary) of this area is 5 units, while the inner boundary starts just after 2 units from the origin. The annular sector is the area between these boundaries.

Step 3: Identifying the Angular Sector

The angular sector determined by \(3\pi/4 < \theta < 5\pi/4\) specifies the part of the plane between these two angles, coming out in a counter-clockwise direction from the positive x-axis. This portion lies partly in the second quadrant and extends into a small part of the third quadrant of the Cartesian plane.

Step 4: Sketching the Region

Draw the polar grid and mark the circles corresponding to \(r = 2\) and \(r = 5\). Highlight the section between these two circles, specifically between the lines \(\theta = 3\pi/4\) (approximately 135 degrees) and \(\theta = 5\pi/4\) (approximately 225 degrees). This shaded area is the region satisfying the given conditions.

Key Concepts

Region Sketching

To sketch a region defined by polar coordinates, it is important to first understand how polar coordinates work. In polar coordinates, each point on the plane is determined by a radius and an angle. The radius (\(r\)) tells us how far the point is from the origin, while the angle (\(\theta\)) tells us the direction of that point from the positive x-axis. In this particular exercise, you are asked to find the set of points where \(2 < r \leq 5\) and \(3\pi/4 < \theta < 5\pi/4\). The goal is to visualize the area on a polar coordinate system by:

• Drawing circles for the provided radii, \(r = 2\) and \(r = 5\).
• Marking angular boundaries at the given angles.
• Identifying and shading the annular region between the two circles which falls within the specified angle range.

Grasping the step-by-step connection of radius and angle in this manner helps simplify the complex idea of a polar coordinate region.

Circular Annular Sector

A circular annular sector is a sliced region of the plane, shaped like a ring or doughnut segment with two concentric circles as its boundaries. This specific sector in the exercise is defined by the radii \(r = 2\) and \(r = 5\), which means:

• The space begins just after the circle of radius 2, extending to the boundary of the circle of radius 5.
• It is bounded by these two circular arcs.

To visualize, imagine a flat hoop of paper, with an imaginary segment cut by two radii. This cut area is the region known as an annular sector. In this exercise, the part between these two circles is the main segment we need to focus on.

Angular Sector

An angular sector in polar coordinates represents a section of the plane divided by two radii lines extending from the origin. In the exercise, the angular sector is defined by angles \(\theta = 3\pi/4\) and \(\theta = 5\pi/4\). This translates the angles to about 135 degrees and 225 degrees, respectively.

• The sector lies primarily within the second quadrant, with part extending into the third quadrant.
• This means that the sector starts at 135 degrees, sweeping through to 225 degrees in a counter-clockwise direction from the positive x-axis.

This area forms a slice of the pie, guiding which section of the circular annular sector to highlight. It is key to note that these angular settings help outline exactly where the annular region should lie on the polar coordinate grid.