Problem 33 - Essential Calculus

Question

Sketch the strophoid \(r=\sec \theta-2 \cos \theta\) \(-\frac{\pi}{2}<\theta<\frac{\pi}{2}\). Convert this equation to rectangular coordinates. Find the area enclosed by the loop.

Step-by-Step Solution

Answer

The graph depicts a strophoid. The rectangular form of the equation can be obtained via the substitution of \(r = \sec \theta - 2\cos\theta\) into the conversion equations. The area within the loop can be determined by integrating \(\frac{1}{2}\int_{a}^{b} r^2 d\theta\) over the appropriate interval for \(\theta\).

Step 1: Sketch the polar equation

The polar equation \(r=\sec \theta-2 \cos \theta\), for \(-\frac{\pi}{2}<\theta<\frac{\pi}{2}\), describes a strophoid or looped shape. Use a graphing calculator or software to visualize it.

Step 2: Convert to Rectangular Coordinates

To transition from polar coordinates to rectangular coordinates, use the conversion equations \(r^2 = x^2 + y^2\) and \(\tan \theta = \frac{y}{x}\). Substitute \(r = \sec \theta - 2 \cos \theta\) and express \(x\) and \(y\) in terms of \(\theta\). Hence \(x = r \cos \theta = (\sec \theta - 2\cos\theta)\cos\theta\) and \(y = r \sin \theta = (\sec \theta - 2\cos\theta)\sin\theta\).

Step 3: Find the Area Enclosed by the Loop

In polar coordinates, the area \(A\) enclosed by the curve is given by the integral \(\frac{1}{2}\int_{a}^{b} r^2 d\theta\). Boundaries for this integral can be determined from the symmetries of the strophoid diagram. Once the area can be expressed as a definite integral, solve this integral to find the area enclosed by the strophoid.

Key Concepts

Polar Coordinates

Polar coordinates are a crucial way of representing points on a plane, especially when dealing with curves like a strophoid. Unlike the usual Cartesian system (using x and y coordinates), polar coordinates locate a point based on its distance from a reference point, known as the pole (similar to the origin in rectangular coordinates), and the angle \( \theta \) from a reference direction. So, a point is expressed as \( r, \theta \), where \( r \) is the distance from the pole, and \( \theta \) is the angle.

When sketching the polar equation \( r = \sec \theta - 2 \cos \theta \), it's important to note that \( \sec \theta \) (the reciprocal of \( \cos \theta \)) leads to some interesting behavior, especially when \( \theta \) approaches \( \pm \frac{\pi}{2} \).

Visualizing similar curves using graphing technology can provide insight into their properties, such as loops, that are often not easily observed on a text alone. With the particular function provided, it’s key to focus on how \( \r\ \) behaves as \( \theta \) changes, leading to the familiar strophoid shape.

Rectangular Coordinates

Rectangular coordinates, or Cartesian coordinates, present another way to define points in a plane using x and y values. To convert polar to rectangular coordinates, we use formulas \( x = r \cos \theta \) and \( y = r \sin \theta \). This method makes it easier to analyze and work with curves algebraically.

For the strophoid \( r = \sec \theta - 2 \cos \theta \), converting involves expressing \( r\ \) as a function of \( \theta\ \), and then substituting this into the equations for \( x\ \) and \( y\ \). Essentially, you'd use:

• \( x = (\sec \theta - 2\cos\theta)\cos\theta \)
• \( y = (\sec \theta - 2\cos\theta)\sin\theta \)

By obtaining the relationships between x and y, we provide a clearer picture of the function in the coordinate plane. This sometimes involves algebraic manipulation to eliminate \( \theta \) and express everything purely in terms of x and y. Such transformations are essential for finding intersections or deriving further properties of the curve.

Area Integral

Finding the area enclosed by a curve is a common application of integral calculus, and it's particularly interesting when using polar coordinates. To compute the area enclosed by a polar curve, one can use the integral formula: \( A = \frac{1}{2}\int_{a}^{b} r^2 d\theta \).

This formula sums up the infinitesimally small sectors of the circle as the angle \( \theta \) varies between two points. It mimics the idea of pie slices stretching out from the pole \( (0, 0) \).

For the equation \( r = \sec \theta - 2 \cos \theta \), determine the appropriate bounds \( a \) and \( b \) by analyzing the symmetry and behavior of the function. Once the integral is correctly set up, evaluating it provides the total area enclosed by the loop of the strophoid. In cases where manual computation is complex, computational tools can assist in finding the numerical area value efficiently.