Problem 11 - Essential Calculus

Question

In Exercises 11 and 12, use a graphing utility to graph the polar equation and find the area of the given region. Inner loop of \(r=1+2 \cos \theta\)

Step-by-Step Solution

Answer

After calculating the integral in step 3, the final answer for the area within the inner loop of \(r = 1+2cos(\theta)\) is found to be \(\pi\) square units.

Step 1: Graphing the Polar Curve

Using a graphing utility or tool like Desmos or Geogebra, plot the polar curve represented by \(r=1+2 \cos \theta\). This polar equation reveals a limaçon, which is a kind of spiralled curve, with an inner loop due to the nature of the cosine function and the coefficients.

Step 2: Identifying the Inner Loop

From the graph, observe that the inner loop of the curve lies within the range \(-\pi/2 \leq \theta \leq \pi/2\). The loop starts form the pole (origin), spirals out and then back in to the pole again over this interval of \(\theta\).

Step 3: Calculating the Area

To find the area (A) inside the inner loop of our curve, we integrate half the square of our radius from \(-\pi/2\) to \(\pi/2\). Mathematically, this is represented as: \(A = \frac{1}{2}\int_{-\pi/2}^{\pi/2} (1+2cos(\theta))^2 d\theta\). Evaluate this definite integral to get the area.

Key Concepts

Graphing Polar Equations

When graphing polar equations, it's essential to understand the system that differs from the Cartesian coordinate system. Polar coordinates are defined with a distance from the origin, known as the radius \( r \), and an angle \( \theta \) from a reference direction, typically the positive x-axis. This is different from (x, y) coordinates where positions are specified with horizontal and vertical distances.

To graph a polar equation like \( r=1+2 \cos \theta \), you can use graphing utilities such as Desmos or Geogebra that support polar plotting. Input the equation and observe how each value of \( \theta \) affects \( r \).

Some key points to remember while graphing polar equations include:

• The effect of trigonometric functions on the shape, rotation, and symmetry of the graph.
• The range of \( \theta \), often spanning from \( 0 \) to \( 2\pi \) for a complete graph.
• Special features such as loops, symmetry, and intersections that can arise due to the equation's terms.

For our specific case, the graph of \( r=1+2 \cos \theta \) forms a limaçon curve with a noticeable inner loop.

Integrals in Polar Coordinates

Integrating in polar coordinates allows for more efficient calculation of areas within polar curves. The integration process involves angles and radii as opposed to the more commonly used x and y axes in Cartesian coordinates.

When calculating areas like in our exercise, we use the formula:
\[ \text{Area} (A) = \frac{1}{2} \int_{a}^{b} (r(\theta))^2 d\theta \]

The integration bounds \( a \) and \( b \) determine the portion of the polar curve to be enclosed. Here \( r(\theta) \) is the polar function, which is squared in the integrand to account for the radial distances. This formula arises because polar integration inherently includes a "sector" of a circle, hence the \( \frac{1}{2} \) coefficient.

It is crucial:

• To correctly identify the limits of integration that correspond to the region of interest.
• To properly square the radius function within the integral.
• To ensure the integration method accounts for trigonometric functions, requiring potential use of trigonometric identities.

The exercise involves using this principle to integrate \( \frac{1}{2}\int_{-\pi/2}^{\pi/2} (1+2\cos(\theta))^2 d\theta \), finding the exact area of the limaçon's inner loop.

Limaçon Curve

A limaçon curve is a type of polar graph described by equations of the form \( r = a + b \cos \theta \) or \( r = a + b \sin \theta \). Depending on the values of \( a \) and \( b \), a limaçon can take on different shapes, some with inner loops and others without.

In the exercise, the equation \( r = 1 + 2 \cos \theta \) is a limaçon with an inner loop, due to the larger coefficient on \( \cos \theta \) compared to the constant \( a \). Here are the typical characteristics of such curves:

• They can have either an inner loop, dimpled form, or no inner loop depending on the ratio of \( a \) to \( b \).
• The inner loop occurs when \( b > a \), like in our equation where 2 is greater than 1.
• The limaçon typically starts at the pole, loops outwards, and may loop back in, depending on \( \theta \).

Understanding these features helps in visualizing the graph, predicting its shape, and properly setting up integral bounds to calculate areas accurately. The specific limaçon curve in the exercise provides an excellent example of a classic looped form familiar in polar coordinate studies.