Problem 30 - Calculus Early Transcendentals: Pearson New International Edition

Question

Sketch the graph of the strophoid \(r=\sec \theta-2 \cos \theta\), and find the area of its loop.

Step-by-Step Solution

Answer

The loop of the strophoid graph is symmetric and its area is calculated by integration in polar coordinates.

Step 1: Understand the Polar Equation

The given equation is in polar coordinates: \( r = \sec \theta - 2 \cos \theta \). To begin, remember that \( \sec \theta = \frac{1}{\cos \theta} \). Thus, the equation becomes \( r = \frac{1}{\cos \theta} - 2 \cos \theta \).

Step 2: Find Cartesian Form

Convert the polar equation to Cartesian coordinates using \( x = r \cos \theta \) and \( y = r \sin \theta \). Start by expressing \( r \) in terms of \( x \) and \( y \) using the trigonometric identities. This step isn't strictly necessary to sketch the graph, but it can aid in understanding the curve.

Step 3: Sketch the Graph of the Strophoid

To sketch the strophoid, identify key points and symmetry. Since \( \theta = \frac{\pi}{2} \) corresponds to an asymptote, and \( r = 0 \) when \( \theta = \pi, 0 \) (for the loop), plot these points first. Consider critical angles to obtain additional points that define the shape of the strophoid.

Step 4: Calculate the Loop Area

The formula to find the area enclosed by a loop in polar coordinates is \( \frac{1}{2} \int_{\alpha}^{\beta} r^2 \, d\theta \). For the loop of the strophoid, set \( \theta \) bounds where the loop occurs, likely within \((\pi, 2\pi)\). Substitute \( r = \sec \theta - 2 \cos \theta \) and solve the integral.

Step 5: Evaluate the Integral

Evaluate the integral \( \frac{1}{2} \int_{\pi}^{2\pi} \left( \frac{1}{\cos \theta} - 2 \cos \theta \right)^2 \, d\theta \). Simplify the expression and compute the definite integral, solving for the area of the loop.

Key Concepts

Polar Coordinates

Polar coordinates are a two-dimensional coordinate system where each point on a plane is determined by a distance from a reference point and an angle from a reference direction. Unlike Cartesian coordinates which use an x-axis and a y-axis, polar coordinates use:

• r: the radial coordinate, representing the distance from the origin.
• θ (theta): the angular coordinate, representing the angle in radians from the positive x-axis.

In the case of the strophoid equation, \( r = \sec \theta - 2 \cos \theta \), these two components define the shape by specifying points as you move through various angles \( \theta \). By calculating \( r \) for different values of \( \theta \) from 0 to \(2\pi\), you can trace the full curve.
For more complex curves like the strophoid, polar coordinates sometimes reveal symmetrical properties or special features (like loops or asymptotes) that might not be easily observed in Cartesian coordinates.

Polar to Cartesian Conversion

Converting polar coordinates to Cartesian coordinates can help visualize complex equations. In such conversions, we utilize trigonometric identities and the relationship between the two systems:

• x, y components: The Cartesian form can express the point using the formulas: \( x = r \cos \theta \) and \( y = r \sin \theta \).
• Purpose of conversion: It simplifies mathematical operations and visualizations by projecting points onto the familiar Cartesian plane.

For the strophoid represented by \( r = \sec \theta - 2 \cos \theta \), you would start by breaking down the polar terms:- Substitute \( r \cos \theta = 1 - 2 \cos^2 \theta \)- Then relate to \( x \) and \( y \) using the above relationships.
While it’s not always necessary for sketching every polar curve, the conversion can help affirm certain geometrical characteristics and facilitate easier computations.

Loop Area Calculation

Calculating the area of a loop in a polar graph involves integrating the square of the radial distance over the specified angular interval. The general formula to find the area enclosed by a polar curve is:- \( A = \frac{1}{2} \int_{\alpha}^{\beta} r^2 \, d\theta \)
For the loop in the strophoid \( r = \sec \theta - 2 \cos \theta \), you'll first determine the angles where the loop occurs. For this example, the loop could occur as \( \theta \) goes from \( \pi \) to \( 2\pi \).
Substitute the equation into the area formula:\[ A = \frac{1}{2} \int_{\pi}^{2\pi} \left( \frac{1}{\cos \theta} - 2 \cos \theta \right)^2 \, d\theta \]Evaluate this definite integral helps give the area of the loop.

• Set up the integrand \( r^2 \) to clarify the calculations.
• Adjust the limits of integration to capture just the desired loop section.

By solving the integral, you can precisely determine the region enclosed by the loop within the graph of the strophoid.