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Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblem 12.3.73
Chapter 12, Problem 12.3.73

63–74. Arc length of polar curves Find the length of the following polar curves.


{Use of Tech} The complete limaçon r=4−2cosθ

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Recall the formula for the arc length \( L \) of a polar curve \( r = r(\theta) \) from \( \theta = a \) to \( \theta = b \): \[ L = \int_{a}^{b} \sqrt{r(\theta)^2 + \left(\frac{d r}{d \theta}\right)^2} \, d\theta \]
Identify the given polar curve: \( r = 4 - 2 \cos \theta \). Since the problem asks for the complete limaçon, the interval for \( \theta \) is from \( 0 \) to \( 2\pi \).
Compute the derivative \( \frac{d r}{d \theta} \) with respect to \( \theta \): \[ \frac{d r}{d \theta} = \frac{d}{d \theta} (4 - 2 \cos \theta) = 2 \sin \theta \]
Substitute \( r(\theta) \) and \( \frac{d r}{d \theta} \) into the arc length formula: \[ L = \int_{0}^{2\pi} \sqrt{(4 - 2 \cos \theta)^2 + (2 \sin \theta)^2} \, d\theta \]
Simplify the expression inside the square root as much as possible before integrating, then evaluate the integral over \( 0 \leq \theta \leq 2\pi \) to find the total arc length.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Polar Coordinates and Polar Curves

Polar coordinates represent points in the plane using a radius and an angle (r, θ). Polar curves are defined by equations relating r and θ, such as r = 4 - 2cosθ. Understanding how to interpret and plot these curves is essential for analyzing their properties, including arc length.
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Intro to Polar Coordinates

Arc Length Formula for Polar Curves

The arc length of a polar curve r(θ) from θ = a to θ = b is given by the integral ∫ₐᵇ √[r(θ)² + (dr/dθ)²] dθ. This formula combines the radius and its rate of change to measure the curve's length accurately. Applying this formula requires computing the derivative dr/dθ and evaluating the integral over the specified interval.
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Arc Length of Parametric Curves

Limaçon Curves and Their Properties

A limaçon is a type of polar curve characterized by equations like r = a + b cosθ or r = a + b sinθ. Depending on parameters, it can have loops or dimpled shapes. Recognizing the limaçon form helps anticipate the curve's behavior and the interval for θ when calculating the complete arc length.
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Related Practice
Textbook Question

15–22. Sets in polar coordinates Sketch the following sets of points.


2 ≤ r ≤ 8

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Textbook Question

13–30. Graphing conic sections Determine whether the following equations describe a parabola, an ellipse, or a hyperbola, and then sketch a graph of the curve. For each parabola, specify the location of the focus and the equation of the directrix; for each ellipse, label the coordinates of the vertices and foci, and find the lengths of the major and minor axes; for each hyperbola, label the coordinates of the vertices and foci, and find the equations of the asymptotes.


4x = -y²

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Textbook Question

31–36. Eliminating the parameter Eliminate the parameter to express the following parametric equations as a single equation in x and y.


x=2 sin 8t, y=2 cos 8t 

Textbook Question

13–30. Graphing conic sections Determine whether the following equations describe a parabola, an ellipse, or a hyperbola, and then sketch a graph of the curve. For each parabola, specify the location of the focus and the equation of the directrix; for each ellipse, label the coordinates of the vertices and foci, and find the lengths of the major and minor axes; for each hyperbola, label the coordinates of the vertices and foci, and find the equations of the asymptotes.


x² = 12y

Textbook Question

39–50. Equations of ellipses and hyperbolas Find an equation of the following ellipses and hyperbolas, assuming the center is at the origin. 

An ellipse with vertices (±5, 0), passing through the point (4, 3/5)

Textbook Question

15–30. Working with parametric equations Consider the following parametric equations.

a. Eliminate the parameter to obtain an equation in x and y.

b. Describe the curve and indicate the positive orientation.


x = cos t, y = sin² t; 0 ≤ t ≤ π

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