Textbook Question
44–49. Areas of regions Find the area of the following regions.
The region inside the cardioid r=1+cosθ and outside the cardioid r=1−cosθ
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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.R.12a
10–12. Parametric curves
a. Eliminate the parameter to obtain an equation in x and y.
x = ln t, y = 8ln t², for 1 ≤ t ≤ e²; (1, 16)
Verified step by step guidance1
Start with the given parametric equations: \(x = \ln t\) and \(y = 8 \ln t^{2}\), where \(1 \leq t \leq e^{2}\).
Recall the logarithm property: \(\ln t^{2} = 2 \ln t\). Use this to rewrite \(y\) as \(y = 8 \times 2 \ln t = 16 \ln t\).
Since \(x = \ln t\), substitute \(\ln t\) in the expression for \(y\) to get \(y = 16x\).
This gives the Cartesian equation relating \(x\) and \(y\) without the parameter \(t\): \(y = 16x\).
Note the domain for \(t\) translates to \(x\) because \(x = \ln t\). Since \(1 \leq t \leq e^{2}\), then \(0 \leq x \leq 2\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Parametric Equations
Parametric equations express the coordinates of points on a curve as functions of a parameter, usually denoted t. Instead of y as a function of x, both x and y depend on t, allowing the description of more complex curves and motions.
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Guided course
08:02
Parameterizing Equations
Eliminating the Parameter
Eliminating the parameter involves manipulating the parametric equations to remove the parameter t, resulting in a direct relationship between x and y. This often requires solving one equation for t and substituting into the other to find an explicit or implicit equation in x and y.
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Guided course
05:59Eliminating the Parameter
Logarithmic Functions and Their Properties
Logarithmic functions, like ln(t), are the inverses of exponential functions and have properties such as ln(a^b) = b ln(a). Understanding these properties is essential for simplifying expressions and eliminating parameters when the parametric equations involve logarithms.
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06:21Properties of Functions
Related Practice
Textbook Question
7–8. Parametric curves and tangent lines
a. Eliminate the parameter to obtain an equation in x and y.
x = 4sin 2t, y = 3cos 2t, for 0 ≤ t ≤ π; t = π/6
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Textbook Question
53–57. Conic sections
b. Use analytical methods to determine the location of the foci, vertices, and directrices.
x² - y²/2 = 1
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Textbook Question
Conic parameters: A hyperbola has eccentricity e = 2 and foci (0, ±2). Find the location of the vertices and directrices.
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Textbook Question
10–12. Parametric curves
a. Eliminate the parameter to obtain an equation in x and y.
x = 3cos(-t), y = 3sin(-t) - 1, for 0 ≤ t ≤ π; (0, -4)
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Textbook Question
61–64. Polar equations for conic sections Graph the following conic sections, labeling vertices, foci, directrices, and asymptotes (if they exist). Give the eccentricity of the curve. Use a graphing utility to check your work.
r = 3/(1 - 2 cos θ)
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