Problem 27 - Thomas Calculus

Question

Centroid of a cardioid region Find the centroid of the region enclosed by the cardioid \(r=1+\cos \theta .\)

Step-by-Step Solution

Answer

The centroid of the cardioid is \( \left(\frac{3}{8}, 0\right) \).

Step 1: Understand the Cardioid Region

The given cardioid is described in polar coordinates as \( r = 1 + \cos \theta \). The cardioid is symmetric with respect to the x-axis and has a cusp at the pole (origin). The range of \( \theta \) is from \( 0 \) to \( 2\pi \).

Step 2: Convert to Cartesian Coordinates for Clarity

The polar equations convert to Cartesian as: \( x = r \cos(\theta) = (1 + \cos \theta) \cos \theta \) and \( y = r \sin(\theta) = (1 + \cos \theta) \sin \theta \). This helps visualize the region we're working with.

Step 3: Set Up the Area Integral

The area \( A \) of the cardioid can be found using the integral \( A = \frac{1}{2} \int_{0}^{2\pi} (r^2) \,d\theta \), where \( r = 1 + \cos \theta \).

Step 4: Calculate the Area

Substitute the given \( r = 1 + \cos \theta \) and solve: \[ A = \frac{1}{2} \int_{0}^{2\pi} (1+\cos \theta)^2 \,d\theta \]. This evaluates to \( A = \frac{3\pi}{2} \).

Step 5: Determine Centroid Coordinates Formulas

The centroid \((\overline{x}, \overline{y})\) in polar coordinates is \( \overline{x} = \frac{1}{A} \int_{0}^{2\pi} \frac{r^2 \cos \theta}{2} \, d\theta \) and \( \overline{y} = \frac{1}{A} \int_{0}^{2\pi} \frac{r^2 \sin \theta}{2} \, d\theta \).

Step 6: Calculate \( \overline{x} \)

Substitute the expressions into the formula for \( \overline{x} \): \[ \overline{x} = \frac{1}{\frac{3\pi}{2}} \int_{0}^{2\pi} \frac{(1+\cos \theta)^2 \cos \theta}{2} \,d\theta \]. Solving this, we find \( \overline{x} = \frac{3}{8} \).

Step 7: Calculate \( \overline{y} \)

Since the cardioid is symmetric about the x-axis, \( \overline{y} = 0 \). The integral to solve would have been \( \overline{y} = \frac{1}{\frac{3\pi}{2}} \int_{0}^{2\pi} \frac{(1+\cos \theta)^2 \sin \theta}{2} \,d\theta \). Due to symmetry, this evaluates to zero.

Key Concepts

Polar Coordinates

Polar coordinates provide a unique way to locate points on a plane using a radius and an angle. This system is especially useful when dealing with curves like cardioids, where symmetry around a central point is a feature.
When using polar coordinates, a point is represented as \( (r, \theta) \), where \( r \) is the distance from the origin, and \( \theta \) is the angle from the positive x-axis.

• The cardioid described by \( r = 1 + \cos \theta \) is a perfect example of how polar coordinates can simplify closed curve descriptions.
• The cardioid's property of having a cusp at the origin highlights how the simplicity of polar coordinates eases the manipulation of complex shapes.

When converting to Cartesian coordinates, formulas like \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \) are used to visualize this shape in the more familiar rectangular system.

Cartesian Coordinates

Switching to Cartesian coordinates can help visualize complicated regions by using the more familiar \( x \) and \( y \) axes.

• In our example, converting the cardioid from polar to Cartesian coordinates involves applying the equations: \( x = (1 + \cos \theta) \cos \theta \) and \( y = (1 + \cos \theta) \sin \theta \).
• This conversion can assist in identifying key features such as symmetry and intercepts with the axes.

Visualizing the cardioid using Cartesian coordinates allows one to perceive the area and centroid processes intuitively, often confirming the simpler interpretations offered by polar descriptions.

Area of a Region

Finding the area of a region in polar coordinates involves integrating over the squared radius across the angle variable. For a cardioid like \( r = 1 + \cos \theta \), the formula transforms to \(A = \frac{1}{2} \int_{0}^{2\pi} (1 + \cos \theta)^2 \,d\theta \).
Using this integral calculus technique, we determine the precise area enclosed by the cardioid to be \( \frac{3\pi}{2} \).

• Integrals serve as powerful tools to calculate complex areas which might be challenging to compute using basic geometry.
• This specific integral accounts for the continuous and curved nature of the cardioid, reinforcing how integration is indispensable in obtaining exact areas.

Symmetry in Geometry

Symmetry plays a central role in simplifying calculations such as finding centroids. Symmetric figures, like the given cardioid, allow predictions about their centroids' coordinates positioning.
The cardioid \( r = 1 + \cos \theta \) is symmetric about the x-axis, and this impacts centroid calculations, especially for the \( \overline{y} \) coordinate.

• In this case, symmetry allows us to immediately conclude that \( \overline{y} = 0 \) without performing complex integrals.
• This insight underscores symmetry's utility in reducing the computational workload—identifying symmetrical properties shortcuts processes in geometry and calculus.

Understanding symmetry helps in recognizing patterns and simplifying problems which otherwise might seem daunting to solve.