Problem 28 - University Calculus: Early Transcendentals

Question

Find the area of the region that lies inside the cardioid \(r=1+\cos \theta\) and outside the circle \(r=1.\)

Step-by-Step Solution

Answer

The area is \(\frac{\pi}{4}\).

Step 1: Understand the Problem

We are asked to find the area of the region that lies inside the cardioid defined by the polar equation \(r = 1 + \cos \theta\) and outside the circle given by \(r = 1\). The goal is to find the overlapping region between these two curves.

Step 2: Find Intersection Points of the Curves

To find where the cardioid and the circle intersect, equate the polar equations: \[1 + \cos \theta = 1.\] Subtract 1 from both sides to obtain:\[\cos \theta = 0.\] The solutions to \(\cos \theta = 0\) are \(\theta = \frac{\pi}{2}\) and \(\theta = \frac{3\pi}{2}\). These are the points of intersection.

Step 3: Set Up the Integral for the Area

The area inside the cardioid from \(\theta = \frac{\pi}{2}\) to \(\theta = \frac{3\pi}{2}\) and outside the circle is computed by evaluating:\[\text{Area} = \frac{1}{2} \int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} ((1 + \cos \theta)^2 - 1^2) \, d\theta.\] Here, \((1 + \cos \theta)^2\) is the square of the cardioid's radius and \(1^2\) is the square of the circle's radius.

Step 4: Simplify the Integrand

Simplify \((1 + \cos \theta)^2 - 1\):\[(1 + \cos \theta)^2 = 1 + 2\cos \theta + \cos^2 \theta.\]Hence,\[(1 + \cos \theta)^2 - 1 = 2\cos \theta + \cos^2 \theta.\] Substitute this back into the integral.

Step 5: Evaluate the Integral

Calculate the integral:\[\text{Area} = \frac{1}{2} \int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} (2\cos \theta + \cos^2 \theta) \, d\theta.\]Break it into simpler terms:\[= \frac{1}{2} \left( 2\int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} \cos \theta \, d\theta + \int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} \cos^2 \theta \, d\theta \right).\]Integrate each part separately.

Step 6: Calculate Each Integral Separately

First integral: \(2\int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} \cos \theta \, d\theta = 0\), as \(\cos \theta\) is symmetric and its integral over a full period is zero.Second integral requires using the identity \(\cos^2 \theta = \frac{1 + \cos(2\theta)}{2}\), leading to:\[\int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} \cos^2 \theta \, d\theta = \frac{1}{2} \int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} (1 + \cos(2\theta)) \, d\theta.\]Evaluating this gives:\[\frac{1}{2} \left[ \theta + \frac{1}{2} \sin(2\theta) \right]_{\frac{\pi}{2}}^{\frac{3\pi}{2}} = \frac{\pi}{2}.\]

Step 7: Compute the Total Area

Combine the results of both integrals:\[\text{Area} = \frac{1}{2} \times \frac{\pi}{2} = \frac{\pi}{4}.\]Thus, after evaluating the integrals, the area of the region bounded by these curves is \(\frac{\pi}{4}\).

Key Concepts

Cardioid

The cardioid is a unique type of curve that we often encounter in polar coordinates. Its shape is similar to that of a heart, hence the name "cardioid," from the Greek word for "heart." A cardioid can be described by the polar equation \( r = 1 + \cos \theta \). This equation plays a significant role in defining the boundary for certain regions in mathematics, particularly when dealing with areas in polar coordinates. The radius \( r \) varies depending on the angle \( \theta \), with the largest radius occurring when \( \cos \theta \) is at its maximum, which is 1.
The cardioid touches the origin (the point at \( (0,0) \)) and loops back around, creating an entire closed shape. Understanding the structure of a cardioid is crucial for solving problems involving areas in polar coordinates, as it determines how we integrate to find these areas.

Intersection Points

Finding intersection points where two curves meet in polar coordinates is essential for defining limits on integration. For our problem involving a cardioid \( r = 1 + \cos \theta \) and a circle \( r = 1 \), we find the points by setting the equations equal to one another: \( 1 + \cos \theta = 1 \).
Simplifying this gives us \( \cos \theta = 0 \). The solutions \( \theta = \frac{\pi}{2} \) and \( \theta = \frac{3\pi}{2} \) provide our points of intersection.

• These points are critical as they represent the angles where the two curves overlap.
• They help determine the limits of integration for the definite integral used to find the area.

Mastery of finding intersection points in polar coordinates will support solving a wide variety of area problems.

Definite Integral

The process of calculating areas in polar coordinates often utilizes the definite integral. In this exercise, our goal is to determine the area inside the cardioid \( r = 1 + \cos \theta \) but outside the circle \( r = 1 \).
This requirement turns into finding the definite integral:\[ \text{Area} = \frac{1}{2} \int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} \left((1 + \cos \theta)^2 - 1^2\right) \, d\theta. \]
The integrand is derived from subtracting the squared equation of the circle from that of the cardioid, giving us the difference in the areas.

• The fraction \( \frac{1}{2} \) in front of the integral accounts for the conversion from rectangular to polar form.
• The limits of the integral, from \( \frac{\pi}{2} \) to \( \frac{3\pi}{2} \), are established by our intersection points, ensuring we only calculate the overlapping region.

Understanding how definite integrals work within polar coordinates is vital for efficiently solving such area problems.

Trigonometric Identities

Trigonometric identities are pivotal when simplifying integrals, especially in polar coordinate geometry. For our problem, simplifying the integrand involves using identities like \( \cos^2 \theta = \frac{1 + \cos(2\theta)}{2} \). This identity assists in handling the square of the cosine function, making the integration process more manageable.
The problem becomes more tractable when broken down using these identities:

• They convert complex expressions into simpler ones by expressing them in terms of different trigonometric functions.
• For example, separating \( 2\cos \theta + \cos^2 \theta \) facilitates the integration into two simpler parts.

Trigonometric identities are fundamental tools in calculus that allow deeper exploration and solution of problems involving periodic functions like sine and cosine.