Problem 34 - Thomas Calculus

Question

Average value If \(f\) is continuous, the average value of the polar coordinate \(r\) over the curve \(r=f(\theta), \alpha \leq \theta \leq \beta,\) with respect to \(\theta\) is given by the formula $$r_{\text { av }}=\frac{1}{\beta-\alpha} \int_{\alpha}^{\beta} f(\theta) d \theta$$ Use this formula to find the average value of \(r\) with respect to \(\theta\) over the following curves \((a>0)\) a. The cardioid \(r=a(1-\cos \theta)\) b. The circle \(r=a\) c. The circle \(r=a \cos \theta, \quad-\pi / 2 \leq \theta \leq \pi / 2\)

Step-by-Step Solution

Answer

a. \( r_{\text{av}} = a \), b. \( r_{\text{av}} = a \), c. \( r_{\text{av}} = \frac{2a}{\pi} \).

Step 1: Understanding the Formula

The average value of a polar coordinate function \( r = f(\theta) \) over an interval \([ \alpha, \beta ]\) is given by \[ r_{\text{av}} = \frac{1}{\beta - \alpha} \int_{\alpha}^{\beta} f(\theta) \, d\theta \]. This formula requires us to integrate the function and then divide by the interval length.

Step 2: Solution for Cardioid

For the cardioid \( r = a(1 - \cos \theta) \), we are asked to find the average value over one full revolution. The typical range for a cardioid is \( \theta = 0 \) to \( \theta = 2\pi \). Substitute \( f(\theta) = a(1 - \cos \theta) \), \( \alpha = 0 \), and \( \beta = 2\pi \) into the formula. Thus, the average value becomes\[ r_{\text{av}} = \frac{1}{2\pi - 0} \int_{0}^{2\pi} a(1 - \cos \theta) \, d\theta. \]Solve the integral: \[ \int_{0}^{2\pi} a(1 - \cos \theta) \, d\theta = a\left[ \theta - \sin \theta \right]_{0}^{2\pi} = a(2\pi - 0) = 2\pi a \].Then, \[ r_{\text{av}} = \frac{1}{2\pi} \cdot 2\pi a = a. \]

Step 3: Solution for Circle with Constant Radius

For the circle \( r = a \), \( f(\theta) = a \) is a constant function. Integrate from \( \theta = 0 \) to \( \theta = 2\pi \):\[ r_{\text{av}} = \frac{1}{2\pi} \int_{0}^{2\pi} a \, d\theta. \]Evaluating the integral gives:\[ \int_{0}^{2\pi} a \, d\theta = a[\theta]_{0}^{2\pi} = a(2\pi) = 2\pi a. \]Therefore, \[ r_{\text{av}} = \frac{2\pi a}{2\pi} = a. \]

Step 4: Solution for Circle with Variable Radius

For the circle \( r = a\cos\theta \), the interval is \( -\pi/2 \leq \theta \leq \pi/2 \). Substitute \( f(\theta) = a\cos\theta \), \( \alpha = -\pi/2 \), \( \beta = \pi/2 \) into the formula:\[ r_{\text{av}} = \frac{1}{\pi} \int_{-\pi/2}^{\pi/2} a\cos\theta \, d\theta. \]Evaluate the integral:\[ \int_{-\pi/2}^{\pi/2} a\cos\theta \, d\theta = a[\sin\theta]_{-\pi/2}^{\pi/2} = a(1 - (-1)) = 2a. \]Thus, \[ r_{\text{av}} = \frac{2a}{\pi}. \]

Key Concepts

Average Value

The average value of a function in polar coordinates helps us understand the typical size of the function over an interval. In polar coordinates, where our function is expressed as \( r = f(\theta) \), we use the formula:

• \( r_{\text{av}} = \frac{1}{\beta - \alpha} \int_{\alpha}^{\beta} f(\theta) \, d\theta \)

This formula integrates the function \( f(\theta) \) over the interval \([\alpha, \beta]\) and then divides by the length of that interval.
The average value gives us a single number that represents the average "height" or value of \( r \) over a curve, helping us get a sense of the overall behavior of the function in that range.
It's particularly useful in shapes like cardioids or circles where understanding their behavior over a full rotation or specific sections is key.

Integration

Integration is a fundamental concept in calculus that allows us to find the total of a quantity, like area under a curve, by summing up small pieces over an interval.
In the context of polar coordinates, integration helps determine the total distance or value around a curve. By setting up an integral from \( \alpha \) to \( \beta \), and considering \( f(\theta) \), we capture all contributions of \( r \) along \( \theta \).

• Integration accumulates values of \( f(\theta) \) over an interval.
• Result helps find the average or total "area" or "length" in polar coordinates.

Using integration in problems with cardioids and circles allows us to compute not only their average values but also understand their properties as entire entities over complete revolutions.
Integration is the tool that puts the average value formula into action by evaluating the necessary total to be averaged.

Cardioid

A cardioid is a heart-shaped curve that can be described in polar coordinates. Its general equation is \( r = a(1 - \cos \theta) \), where \( a \) is a positive constant.
In the context of calculating average values:

• Cardioids trace a complete shape over the interval from \( \theta = 0 \) to \( \theta = 2\pi \).
• To find the average value, we integrate over one full cycle.

The interesting property of a cardioid is that it's symmetrical and has a cusp. It results in an average value of \( r = a \), showcasing that despite its shape, the average distance from the origin (in terms of \( r \)) is still proportional to \( a \).
The cardioid's simplicity makes it a great starting point for understanding more complex polar curves.

Circle

Circles are familiar geometric shapes that can be efficiently described using polar coordinates. In these problems, we often see two forms of circles:

• A constant radius, described by \( r = a \).
• A variable radius, like \( r = a\cos\theta \).

For a circle with constant radius, \( r = a \), the average value across \( \theta = 0 \) to \( 2\pi \) simply equals \( a \). The situation reaffirms that every point on the circle is equally distant from the center.
For a circle where \( r = a\cos\theta \), the average value over \( -\pi/2 \leq \theta \leq \pi/2 \) is \( \frac{2a}{\pi} \). This modification illustrates how the circle's structure, having different maximum and minima radii based on \( \theta \), influences the average value of \( r \).
Understanding these simple yet fundamental polar curves like circles helps us build toward analyzing more complex shapes in polar coordinates.