Problem 18 - Thomas Calculus

Question

The area of the region that lies inside the cardioid curve \(r=\cos \theta+1\) and outside the circle \(r=\cos \theta\) is not $$\frac{1}{2} \int_{0}^{2 \pi}\left[(\cos \theta+1)^{2}-\cos ^{2} \theta\right] d \theta=\pi$$ Why not? What is the area? Give reasons for your answers.

Step-by-Step Solution

Answer

The correct area is \(\pi\), not as originally stated.

Step 1: Understand the Problem

We need to find the area enclosed by the cardioid given by the polar equation \( r = \cos \theta + 1 \) that lies outside the circle given by \( r = \cos \theta \). The area should be computed using polar coordinates.

Step 2: Find the Intersection Points

The intersection of the curves occurs where \( r = \cos \theta + 1 \) is equal to \( r = \cos \theta \). Solving for \( \cos \theta + 1 = \cos \theta \) gives us \( 1 = 0 \), indicating a conceptual mistake. Instead, use \( r = 0 \) for intersection points which gives \( \theta = \pi \) for the circle's explicit intersection with itself.

Step 3: Correct Integral Limits and Setup

To find the correct area, notice symmetry and set the correct integral bounds. For \( \theta \) from \( 0 \) to \( 2\pi \), only values from \( \theta = 0 \) to \( \theta = \pi \) for the region inside the cardioid matter. Calculate from the difference of the cardioid squared minus the circle squared.

Step 4: Calculate the Correct Area Expression

The correct area from the origin to where one checks polar coordinates would be \[\frac{1}{2} \int_0^{\pi} \left((\cos \theta + 1)^2 - \cos^2 \theta\right) \, d\theta.\]Recalculate this for accuracy.

Step 5: Evaluate the Integral

To solve the integral:1. Expand \((\cos \theta + 1)^2 = \cos^2 \theta + 2 \cos \theta + 1\).2. Subtract \( \cos^2 \theta \) yielding \( 2 \cos \theta + 1 \).3. Integrate from \(0\) to \(\pi\).4. Result: \( \int_0^{\pi} (1 + 2 \cos\theta) \, d\theta = \pi \). Verify limits and results.

Key Concepts

area between curves

To determine the area between two curves in polar coordinates, you must assess the regions enclosed by each curve separately. In our scenario, we have a cardioid and a circle. The point is to calculate the area inscribed within the cardioid that doesn't overlap with the circle. In polar form, the area within a curve over a specific angle range can be calculated using this formula: \[A = \frac{1}{2} \int_{\theta_1}^{\theta_2} r^2 \, d\theta\]Using this approach, we need to cover the segments where the cardioid is outside the circle. Adjust the integration bounds as necessary, and compute using the squared difference of the radii functions.

integral calculus

Integral calculus plays a central role in finding areas under curves, not just in Cartesian coordinates but also in polar coordinates. With polar coordinates, the process entails calculating the integral of one-half of `r` squared overspecified angle bounds. Evaluating the integral involves splitting it into manageable components:

• Expand and simplify expressions separately.
• Consider the results between specific limits.
• Ensure accuracy in trigonometric evaluations and symmetry usage.

Relying on integral calculus allows us to derive exact area representations from complex polar curves, like our example's cardioid and circle.

cardioid

A cardioid is a heart-shaped curve represented in polar coordinates by the equation \( r = 1 + \cos \theta \). Its distinct shape results from its formation method—a circle's path rolling around another fixed circle of equal radius.The cardioid displays symmetrical properties about the polar axis and is a valuable study paradigm within calculus due to its tractable mathematical properties. For our exercise, the cardioid serves as the outer boundary within which we calculate the specified area, excluding the arc specified by the circle where they overlap.

intersection points

Finding intersection points is paramount when determining areas bound by multiple curves. These points mark limits of integration and provide key insights into the curves' relative positions.In polar coordinates, intersection points are obtained by equating the `r` values of the involved polar equations. If direct solutions are not evident—as with our problem where faulty logic led to a non-logical conclusion—conceptual thinking and trigonometric insight (like evaluating when `r` equals zero) helps.Identifying \(\theta = \pi\) as a critical intersection leads us to redefine integration limits and simplifications, ensuring efficient area calculations.