Problem 7 - Calculus Early Transcendentals

Question

\(7-12\) Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions. $$1\leqslant r \leqslant 2$$

Step-by-Step Solution

Answer

Sketch the region between circles with radii 1 and 2 centered at the origin.

Step 1: Convert Conditions to Cartesian

Given that the polar coordinates are \(1 \leq r \leq 2\), it implies we are considering all points with these radii. In Cartesian coordinates, this describes a region between two circles centered at the origin, with radii 1 and 2. The region is the annular region between these circles.

Step 2: Sketch Inner Circle

Start by sketching the inner circle with a radius of 1 centered at the origin \((0,0)\). This circle can be described by the equation \(x^2 + y^2 = 1\).

Step 3: Sketch Outer Circle

Next, sketch the outer circle with a radius of 2, also centered at the origin. This circle has the equation \(x^2 + y^2 = 4\).

Step 4: Highlight the Annular Region

The region of interest is the area between the two circles drawn in Steps 2 and 3. This forms an annular region or ring.

Step 5: Review Complete Sketch

Examine the graph to ensure the annular region between the circles is clearly marked without shading inside the inner circle or outside the outer circle.

Key Concepts

Annular Region

An annular region is essentially a donut-shaped area between two concentric circles. Concentric circles share the same center but have different radii. In this case, the annular region is formed by all points located between two circles centered at origin (0,0). The inner circle possesses a smaller radius, while the outer circle has a larger one.

The annular region doesn't include points inside the smaller circle or outside the larger one. You can think of it as the area you get when you remove the smaller circle from the larger one. This concept is very useful in various scientific and engineering disciplines, where combining different shapes helps in modeling real-world phenomena.

Cartesian Coordinates

Cartesian coordinates are a way to represent points in a plane using two numbers, typically denoted as (x, y). This coordinate system uses perpendicular axes to identify any point's location.

For circles and their representations, the center of the circle is located at the origin or (0,0) in Cartesian terms. Polar coordinates, another type of system, would instead use a distance measure (like radius, r) and an angle to define points. Converting between polar and Cartesian coordinates often simplifies solving geometric problems.

• The x-coordinate tells how far to move horizontally from the origin.
• The y-coordinate tells how far to move vertically from the origin.

In the context of our exercise, understanding Cartesian coordinates helps us better interpret the regions defined by polar coordinate conditions. It's like translating the language of circles into a format we can easily depict and analyze on paper.

Circle Equations

Circle equations in the Cartesian coordinate system offer a precise way to describe circles. The standard equation for a circle centered at the origin (0,0) is \(x^2 + y^2 = r^2\), where \(r\) is the radius of the circle.

Understanding circle equations is fundamental to sketching these shapes accurately. In our exercise, the equation \(x^2 + y^2 = 1\) represents a circle with a radius of 1, while \(x^2 + y^2 = 4\) represents a circle with a radius of 2.

These equations help draw each circle precisely and identify the boundaries of our regions of interest. By manipulating these equations, such as changing the radius or the center, we can model different circular shapes for various mathematical and practical applications. These skills are essential in both pure geometry and applied fields.