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Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblema 12.3.76
Capitolo 12, Problema 12.3.76

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


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

Guida verificata passo dopo passo
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: \( r = 4 - 2 \cos \theta \). We need to find \( \frac{d r}{d \theta} \), the derivative of \( r \) with respect to \( \theta \).
Compute the derivative: \[ \frac{d r}{d \theta} = \frac{d}{d \theta} (4 - 2 \cos \theta) = 2 \sin \theta \]
Determine the interval for \( \theta \) to cover the complete limaçon. Since the limaçon is a closed curve 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: \[ L = \int_{0}^{2\pi} \sqrt{(4 - 2 \cos \theta)^2 + (2 \sin \theta)^2} \, d\theta \] This integral can then be evaluated (analytically or using technology) to find the length of the limaçon.

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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 geometric properties.
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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.
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Percorso guidato
06:29
Arc Length of Parametric Curves

Use of Technology in Calculus

Technology such as graphing calculators or computer algebra systems can assist in calculating derivatives, evaluating integrals, and visualizing curves. For complex integrals like the arc length of a limaçon, technology helps obtain precise numerical results efficiently.
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Percorso guidato
06:11
Fundamental Theorem of Calculus Part 1
Pratica correlata
Domanda del libro di testo

90–94. Focal chords A focal chord of a conic section is a line through a focus joining two points of the curve. The latus rectum is the focal chord perpendicular to the major axis of the conic. Prove the following properties.

The length of the latus rectum of a hyperbola centered at the origin is (2b²)/a = 2b√(1 - e²)

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Domanda del libro di testo

37–48. Polar-to-Cartesian coordinates Convert the following equations to Cartesian coordinates. Describe the resulting curve.


r cos θ = -4

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Domanda del libro di testo

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


A hyperbola with vertices (±4, 0) and foci (±6, 0)

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Domanda del libro di testo

57–64. Graphing polar curves Graph the following equations. Use a graphing utility to check your work and produce a final graph.


r² = 4 sin θ  

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Domanda del libro di testo

Air drop—inverse problem A plane traveling horizontally at 100 m/s over flat ground at an elevation of 4000 m must drop an emergency packet on a target on the ground. The trajectory of the packet is given by

x = 100t, y = −4.9t² + 4000, t ≥ 0

where the origin is the point on the ground directly beneath the plane at the moment of the release. How many horizontal meters before the target should the packet be released in order to hit the target?

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Domanda del libro di testo

25–30. Converting coordinates Express the following polar coordinates in Cartesian coordinates.


(4, 5π)

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