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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 12.3.67
Chapter 12, Problem 12.3.67

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


The complete cardioid r = 4 + 4 sin θ

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1
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: \( r = 4 + 4 \sin \theta \). We need to find \( \frac{d r}{d \theta} \), the derivative of \( r \) with respect to \( \theta \).
Compute the derivative: \[ \frac{d r}{d \theta} = 4 \cos \theta \]
Determine the interval for \( \theta \) that traces the complete cardioid. Since cardioids are typically traced once as \( \theta \) goes from \( 0 \) to \( 2\pi \), set the limits of integration as \( a = 0 \) and \( b = 2\pi \).
Set up the integral for the arc length: \[ L = \int_{0}^{2\pi} \sqrt{(4 + 4 \sin \theta)^2 + (4 \cos \theta)^2} \, d\theta \] This integral can then be simplified and evaluated to find the length of the cardioid.

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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 + 4 sin θ. 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 ∫ from a to b of √(r(θ)² + (dr/dθ)²) dθ. This formula accounts for changes in both radius and angle, allowing calculation of the curve's length in polar form.
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Arc Length of Parametric Curves

Differentiation of Polar Functions

To apply the arc length formula, you must differentiate r(θ) with respect to θ. This involves using standard differentiation rules on trigonometric functions like sin θ. Accurate computation of dr/dθ is crucial for evaluating the integral correctly.
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Intro to Polar Coordinates
Related Practice
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 = r − 1, y = r³; −4 ≤ r ≤ 4

Textbook Question

Tangent lines for a hyperbola Find an equation of the line tangent to the hyperbola x²/a² + y²/b² = 1 at the point (x₀, y₀)

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 curve r = √(cos θ)

Textbook Question

Given three polar coordinate representations for the origin.

1
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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 limaçon r = 2 + cos θ

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