Problem 7 - Thomas Calculus

Question

In Exercises \(1-8,\) describe the given region in polar coordinates. The region enclosed by the circle \(x^{2}+y^{2}=2 x\)

Step-by-Step Solution

Answer

The region is described by polar equation \(r = 2 \cos \theta\), \(0 \leq r \leq 2\cos\theta\), \(-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}\).

Step 1: Convert Cartesian Equation to Polar Form

The given circle in Cartesian coordinates is \(x^2 + y^2 = 2x\). Use the relationships between Cartesian and polar coordinates: \(x = r \cos \theta\) and \(y = r \sin \theta\). Substitute these into the equation to get \((r \cos \theta)^2 + (r \sin \theta)^2 = 2r \cos \theta\), which simplifies to \(r^2 = 2r \cos \theta\).

Step 2: Simplify Polar Equation

The expression \(r^2 = 2r \cos \theta\) can be simplified by dividing through by \(r\) (assuming \(r eq 0\)), giving \(r = 2 \cos \theta\). This represents the equation of the circle in polar coordinates.

Step 3: Determine the Range of \(r\) and \(\theta\)

The circle \(r = 2 \cos \theta\) suggests that \(r\) is non-negative (since it represents a radius), and \(\theta\) varies over the range where \(\cos \theta\) is non-negative—between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\). Thus, \(0 \leq r \leq 2 \cos \theta\) and \(\theta\) ranges from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\).

Key Concepts

Cartesian coordinates

Cartesian coordinates are a way of describing points on a plane using pairs of numbers. These coordinates describe each point based on its distance from two perpendicular axes. The horizontal axis is typically called the x-axis, and the vertical axis is the y-axis.

In this coordinate system, any point can be located by specifying a pair of numbers, such as (x, y). Here, the first number represents the distance along the x-axis, and the second number represents the distance along the y-axis:

• The x-coordinate tells you how far left or right the point is from the y-axis.
• The y-coordinate tells you how far up or down the point is from the x-axis.

Using Cartesian coordinates, we can describe geometric shapes, such as circles, by equations that use x and y. For instance, the equation of a circle in Cartesian coordinates might look like this: \(x^2 + y^2 = r^2\), where \(r\) is the radius of the circle.

circle equation

In mathematics, a circle is often described by an equation. The general form of a circle equation in Cartesian coordinates is \(x^2 + y^2 = r^2\), where \(r\) is the circle's radius.

However, circles can also be described in terms of other equations. For example, the given equation \(x^2 + y^2 = 2x\) is not in the standard form. This equation represents a circle that isn't centered at the origin. We can transform such equations into polar coordinates for easier understanding and visualization.

To make sense of equations like this, we often employ techniques to complete the circle equation square or to convert them into more familiar forms.

polar form conversion

Polar coordinates express a point’s location using a distance and an angle rather than two distances as in Cartesian coordinates. In the polar system, a point is defined by \(r\), the radial distance from the origin, and \(\theta\), the angle from the positive x-axis. This format is particularly handy for circular shapes and regions.

To convert from Cartesian coordinates to polar coordinates, you use the following equations:

• \(x = r \cos \theta\)
• \(y = r \sin \theta\)

By substituting \(x\) and \(y\) with these expressions in a Cartesian equation, you can obtain the equivalent polar form. In the original exercise, the circle equation \(x^2 + y^2 = 2x\) was transformed into its polar form \(r = 2 \cos \theta\) by making these substitutions and simplifying the result. This polar equation is more intuitive for identifying circular regions, especially if the circle is centered at the origin.

range determination

Determining the range in polar coordinates involves understanding the limits of \(r\) and \(\theta\).

For the polar equation \(r = 2 \cos \theta\), the range of values \(r\) can take is directly influenced by the behavior of \(\cos \theta\). Since \(\cos \theta\) ranges from -1 to 1:

• \(r\) takes values between 0 and maximum 2 (when \(\cos \theta = 1\)).
• The circle is represented only where \(\cos \theta\) produces a non-negative radial \(r\).

Regarding \(\theta\), this angle ranges from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\). This is because \(\cos \theta\) is non-negative, meaning it is effective only in the first and fourth quadrants of the circle. Hence, in this problem, \(r\) is bounded by \(0 \leq r \leq 2\cos\theta\), and \(\theta\) by \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\). This range restricts the circle's representation to a semicircle oriented along the positive x-direction.