Textbook Question
77–80. Slopes of tangent lines Find all points at which the following curves have the given slope.
x = 2 cos t, y = 8 sin t; slope = -1
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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
How does the eccentricity determine the type of conic section?
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
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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Textbook Question
11–20. Slopes of tangent lines Find the slope of the line tangent to the following polar curves at the given points.
r = 2θ; (π/2, π/4)
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Textbook Question
75–76. Graphs to polar equations Find a polar equation for each conic section. Assume one focus is at the origin.