Textbook Question
57–62. Polar equations for conic sections Graph the following conic sections, labeling the vertices, foci, directrices, and asymptotes (if they exist). Use a graphing utility to check your work.
r = 3/(2 + cos θ)
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Ch.12 - Parametric and Polar Curves
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 12, Problem 12.3.59
45–60. Areas of regions Find the area of the following regions.
The region inside the limaçon r = 4 - 2 cos θ
Verified step by step guidance1
Recall that the area enclosed by a polar curve \( r = f(\theta) \) from \( \theta = a \) to \( \theta = b \) is given by the formula:
\[ A = \frac{1}{2} \int_{a}^{b} r^2 \, d\theta \]
Identify the curve given: \( r = 4 - 2 \cos \theta \). Since this is a limaçon, and the problem asks for the area inside the entire curve, we consider \( \theta \) from 0 to \( 2\pi \) to cover the full region.
Set up the integral for the area using the formula:
\[ A = \frac{1}{2} \int_{0}^{2\pi} (4 - 2 \cos \theta)^2 \, d\theta \]
Expand the square inside the integral to simplify the integrand:
\[ (4 - 2 \cos \theta)^2 = 16 - 16 \cos \theta + 4 \cos^2 \theta \]
Split the integral into separate terms and use known integral formulas for \( \cos \theta \) and \( \cos^2 \theta \) over the interval \( [0, 2\pi] \) to evaluate each part before combining them to find the total area.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Polar Coordinates and Graphing
Polar coordinates represent points using a radius and an angle, expressed as (r, θ). Understanding how to graph polar equations like r = 4 - 2 cos θ helps visualize the region whose area is to be found, especially for curves like limaçons that have distinctive shapes.
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Intro to Polar Coordinates
Area Calculation in Polar Coordinates
The area enclosed by a polar curve r(θ) from θ = a to θ = b is given by (1/2) ∫ from a to b of [r(θ)]² dθ. This formula is essential for finding the area inside curves defined in polar form, requiring setting correct integration limits and squaring the radius function.
Recommended video:
05:32Intro to Polar Coordinates
Properties of Limaçon Curves
Limaçons are a family of polar curves defined by equations like r = a ± b cos θ. Their shape varies depending on the relationship between a and b, which affects whether the curve has an inner loop, dimple, or is convex. Recognizing these properties aids in determining integration bounds and interpreting the region.
Recommended video:
06:21Properties of Functions
Related Practice
Textbook Question
69–72. Tangent lines Find an equation of the line tangent to the following curves at the given point.
y² - x²/64 = 1; (6, -5/4)
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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
31–38. Equations of parabolas Find an equation of the following parabolas. Unless otherwise specified, assume the vertex is at the origin.
A parabola symmetric about the y-axis that passes through the point (2, -6)
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Textbook Question
25–30. Converting coordinates Express the following polar coordinates in Cartesian coordinates.
(1, 2π/3)
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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=t,y= √(4−t²) a
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