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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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Properties of Functions
Related Practice
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.


4x² - y² = 16

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Textbook Question

41–44. Intersection points and area  Find all the intersection points of the following curves. Find the area of the entire region that lies within both curves


r = 1 + sin θ and r = 1 + cos θ

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Textbook Question

45–60. Areas of regions Find the area of the following regions.


The region inside one leaf of the rose r = cos 5θ

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Textbook Question

Golden Gate Bridge Completed in 1937, San Francisco’s Golden Gate Bridge is 2.7 km long and weighs about 890,000 tons. The length of the span between the two central towers is 1280 m; the towers themselves extend 152 m above the roadway. The cables that support the deck of the bridge between the two towers hang in a parabola (see figure). Assuming the origin is midway between the towers on the deck of the bridge, find an equation that describes the cables. How long is a guy wire that hangs vertically from the cables to the roadway 500 m from the center of the bridge? 

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Textbook Question

33–40. Areas of regions Make a sketch of the region and its bounding curves. Find the area of the region.


The region inside the circle r = 8 sin θ

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Textbook Question

63–66. Tracing hyperbolas and parabolas Graph the following equations. Then use arrows and labeled points to indicate how the curve is generated as θ increases from 0 to 2π. 


r = 3/(1 - cos θ)

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